Miles Cranmer - The Next Great Scientific Theory is Hiding Inside a Neural Network (April 3, 2024)

Summary

The transcript discusses the emerging paradigm of interpreting neural networks for physical insights in scientific discovery. It highlights the potential of AI in learning complex models from limited data, exemplified by advances in fluid turbulence and planetary system instability prediction. The speaker emphasizes the importance of translating these models into interpretable mathematical language using symbolic regression. They also introduce the concept of polymathic AI, which involves creating large, flexible neural networks trained on diverse data to serve as foundational models for various scientific tasks, promoting a new approach to building theories in the physical sciences.

Takeaways

• 🧠 The concept of interpreting neural networks for physical insights represents a new paradigm in scientific exploration.
• 🌀 Success in using neural networks for scientific insights includes predicting instability in planetary systems and modeling fluid turbulence with high accuracy.
• 📈 Traditional scientific methods involve building theories from low-dimensional data, while modern AI-driven approaches use high-dimensional data and flexible functions.
• 🔍 The speaker's motivation is to understand how neural networks achieve accuracy and to use these insights to advance scientific understanding.
• 🧬 The potential of machine learning in science is highlighted by the ability of neural networks to learn from data and find patterns not previously recognized.
• 📚 Symbolic regression is a technique used to interpret neural networks by finding analytic expressions that optimize to fit data sets.
• 🧬 The use of genetic algorithms in symbolic regression is akin to evolving equations to fit data, providing a bridge between machine learning models and mathematical language.
• 🌟 Foundation models, like GPT for language, are proposed for science as a way to train on diverse data and then specialize for specific tasks, improving performance.
• 🔄 The concept of 'polymathic AI' is introduced as a foundation model for science that can incorporate data across disciplines and be fine-tuned for particular problems.
• 🔍 The importance of simplicity in scientific models is questioned, with the suggestion that what is considered simple may be based on familiarity and utility rather than inherent simplicity.

Q & A

What is the main motivation behind interpreting neural networks for physical insights?

The main motivation is to extract valuable scientific insights from neural networks, which can potentially advance our understanding of various phenomena and contribute to the development of new theories in the physical sciences.

How does the traditional approach to science differ from the new paradigm of using neural networks?

The traditional approach involves building theories based on low-dimensional data sets or summary statistics, whereas the new paradigm uses massive neural networks to find patterns and insights in large, complex data sets, and then builds theories around what the neural networks have learned.

Can you explain the concept of symbolic regression in the context of interpreting neural networks?

Symbolic regression is a machine learning task that aims to find analytic expressions that optimize some objective by searching over all possible expression trees. It is used to build surrogate models of neural networks, translating the model into a mathematical language that is interpretable and familiar to scientists.

What is the significance of the universal approximation theorem in relation to neural networks?

The universal approximation theorem states that a shallow neural network with a single layer of activations can approximate any 1D function to arbitrary accuracy. This highlights the power of neural networks in modeling complex relationships and functions in data.

How do foundation models like GPT differ from traditional machine learning models?

Foundation models are trained on massive, diverse datasets and are flexible enough to serve as a basis for a wide range of tasks across different domains. They are first pre-trained on general data and then fine-tuned for specific tasks, whereas traditional models are often trained from scratch for a particular task.

What is the role of simplicity in the context of scientific discovery and interpretability?

In the context of scientific discovery, simplicity often refers to the ability to describe complex phenomena with minimal assumptions or variables. It aids interpretability by providing clear, understandable explanations for observed data, which can lead to more effective models and theories.

How does the concept of pre-training neural networks relate to the development of polymathic AI?

Pre-training neural networks on a broad range of data allows them to develop general priors for different types of problems, much like a well-rounded scientist. This approach is central to the development of polymathic AI, which aims to create models that can be fine-tuned for specific tasks across various scientific disciplines.

What are the potential challenges in training a foundation model for science, given the diversity of data types in different scientific fields?

The main challenge lies in defining a general objective that can be applied to the diverse range of data types in science. The objective needs to be flexible enough to accommodate different data forms, such as sequences in molecular biology or images in astrophysics, while still enabling the model to learn broadly applicable concepts.

How does the concept of shared concepts across different physical systems relate to the training of foundation models?

Shared concepts like causality and multiscale dynamics are common across various scientific disciplines. By training a foundation model on diverse datasets that encompass these shared concepts, the model can develop a general understanding of these principles, which can then be fine-tuned for specific tasks within particular fields.

What are the potential implications of polymathic AI for the future of scientific research?

Polymathic AI has the potential to revolutionize scientific research by providing a generalizable foundation model that can quickly adapt to new tasks and problems. This could lead to faster discoveries, more efficient use of computational resources, and the development of new, broadly applicable scientific models.

Outlines

00:00

🤖 Introduction to Neural Network Interpretation

The speaker introduces the concept of interpreting neural networks to gain physical insights, which is seen as a new paradigm in science. The work involves collaboration with many individuals and focuses on extracting insights from neural networks, particularly in the context of fluid turbulence and planetary system instability. The speaker emphasizes the importance of understanding how neural networks achieve accuracy beyond traditional models and the potential for using these insights to advance scientific understanding.

05:03

🧠 Machine Learning Fundamentals and Activation Functions

The speaker delves into the fundamentals of machine learning, starting with linear regression and progressing to the concept of activation functions in neural networks. Activation functions introduce non-linearity, allowing the model to fit complex data. The speaker explains the construction of a shallow neural network and how it can be viewed as a piecewise linear model. The universal approximation theorem is mentioned, highlighting the ability of a neural network to approximate any 1D function to arbitrary accuracy.

10:05

🌀 Deepening Neural Networks and Function Composition

The speaker discusses the progression from shallow to deep neural networks, explaining how additional layers allow for the representation of functions in higher dimensional spaces. The concept of function composition is introduced, likening it to folding a piece of paper to create complex behaviors. The speaker emphasizes the efficiency of neural networks through shared computation and neurons, and how this relates to emulating physical processes and gaining interpretable insights through symbolic regression.

15:06

📝 Symbolic Regression and Surrogate Models

The speaker explains the process of symbolic regression, a machine learning task aimed at finding analytic expressions that fit a dataset. This is done using a genetic algorithm-based search, which evolves equations to fit the data. The speaker discusses the use of symbolic regression to build surrogate models of neural networks, translating the model into a mathematical language. This technique is used to interpret and distill the behavior of neural networks into equations, providing a more interpretable approximation of the original model.

20:07

🌟 Polymathic AI and Foundation Models

The speaker introduces the concept of polymathic AI, focusing on building massive neural networks for science. Foundation models, which are trained on diverse data to learn general concepts, are discussed as a way to improve performance on downstream tasks. The speaker explains how these models can be fine-tuned for specific tasks, drawing parallels with language models. The potential for these models to discover new scientific insights and the importance of pre-training are highlighted, with examples of their application in physics.

25:08

🔍 Interpreting Neural Networks and Simplicity

The speaker discusses the interpretation of neural networks, particularly focusing on the concept of simplicity. The speaker argues that simplicity is based on familiarity and usefulness, and that broadly useful algorithms discovered by polymathic AI models may become familiar and simple over time. The speaker emphasizes the importance of understanding what is interpretable and how pre-training can provide a better starting point for neural networks in scientific applications.

30:10

💡 Final Thoughts and Q&A

The speaker concludes with a discussion on the potential impact of polymathic AI on science and how it might change scientific teaching. The speaker also addresses questions about the scalability of training for foundation models, the potential for these models to discover new physics, and the challenges of symbolic regression with high-dimensional data. The speaker acknowledges that while pre-training has been beneficial in experiments so far, adversarial examples may exist that could hinder training in certain cases.

Keywords

💡Neural Networks

Neural networks are a series of algorithms that attempt to recognize underlying relationships in a set of data through a process that mimics the way the human brain operates. In the context of the video, neural networks are used to interpret and gain physical insights from data, such as learning subgrid models for fluid turbulence or predicting instability in planetary systems.

💡Physical Insight

Physical insight refers to the understanding of the fundamental principles or laws that govern physical phenomena. In the video, the speaker is interested in using neural networks to extract such insights, which can lead to advances in scientific understanding and the development of new theories.

💡Data-Driven Models

Data-driven models are computational models that are developed based on data and statistical analysis, rather than purely on theoretical or physical principles. These models are used to make predictions or identify patterns within large datasets. In the video, the speaker discusses the use of neural networks as data-driven models to emulate physical processes and gain new scientific insights.

💡Symbolic Regression

Symbolic regression is a form of machine learning where the goal is to find an analytical expression that fits a set of data. It is used to create a human-interpretable model from data, which can be expressed in mathematical terms. In the video, the speaker describes using symbolic regression to interpret the behavior of neural networks and translate their learned patterns into mathematical equations.

💡Foundation Models

Foundation models are machine learning models that are trained on massive amounts of general and diverse data to serve as a basis for other models. They are then fine-tuned for specific tasks, often resulting in better performance due to the general priors they have learned. In the video, the speaker discusses the concept of foundation models in the context of scientific discovery and their potential to revolutionize the way science is done.

💡Interpretability

Interpretability in machine learning refers to the ability to understand the reasoning behind a model's predictions or decisions. It is crucial for gaining trust in the model, diagnosing its failures, and leveraging its knowledge. In the video, the speaker emphasizes the importance of interpretability in neural networks to extract meaningful scientific insights and contribute to the advancement of science.

💡Polymathic AI

Polymathic AI refers to a hypothetical type of artificial intelligence that is capable of understanding, learning, and making contributions across multiple disciplines or fields. In the video, the speaker discusses the idea of building such AI by training neural networks on diverse scientific data to create a foundation model that can then be specialized for various scientific tasks.

💡Subgrid Models

Subgrid models are mathematical models used in computational fluid dynamics to represent small-scale processes that occur within the grid cells of a larger simulation. They are essential for accurately modeling turbulence and other complex fluid behaviors. In the video, the speaker mentions subgrid models as an example of how neural networks can be used to improve upon traditional models.

💡Planetary Instability

Planetary instability refers to the condition where the gravitational interactions within a planetary system cause the orbits of the planets to become chaotic and potentially collide or eject each other from the system. In the video, the speaker discusses using neural networks to predict such instabilities, which is a centuries-old problem in the field of chaos theory.

💡Genetic Algorithm

A genetic algorithm is a search heuristic that mimics the process of natural selection to find approximate solutions to optimization and search problems. In the context of the video, genetic algorithms are used for symbolic regression to evolve mathematical expressions that best fit a given dataset.

Highlights

The speaker discusses a new paradigm of interpreting neural networks to gain physical insights, marking a shift in the approach to science.

The motivation stems from examples like using neural networks to learn subgrid models for fluid turbulence with high accuracy, outperforming traditional models.

Another example is predicting instability in planetary systems, a centuries-old problem, where neural networks have shown better accuracy and generalization.

The traditional approach to science involves building theories from low-dimensional data sets, but the speaker suggests using neural networks as a tool to describe the data and its patterns.

Neural networks, especially those trained on massive amounts of data, can discover new things not present in existing theories.

The speaker introduces the concept of using neural networks as a compression tool to extract common patterns from a data set.

The potential of machine learning models to serve as surrogate models for complex physical processes is highlighted.

Symbolic regression is introduced as a method to interpret data-driven models by finding analytic expressions that optimize some objective.

The speaker discusses the use of genetic algorithms for symbolic regression, evolving equations to fit the data set.

Foundation models are presented as a shift in industrial machine learning, which are trained on massive, diverse data and then fine-tuned for specific tasks.

The idea of polymathic AI is introduced, aiming to build massive neural networks for science that incorporate data across many disciplines.

The potential of polymathic AI to discover broadly useful algorithms and new scientific models is emphasized.

The speaker discusses the importance of pre-training models, even on unrelated data, to provide a better starting point than random initialization.

The concept of simplicity is explored, suggesting that what is considered simple is often based on familiarity and usefulness in describing the world.

The speaker argues that polymathic models will be broadly useful and may drive the simplicity of newly discovered concepts or algorithms.

The potential impact of polymathic AI on how science is taught and done is discussed, suggesting a significant shift in the scientific process.

The speaker addresses concerns about whether neural networks are rediscovering known physics or discovering new insights, emphasizing the importance of verification.

The limitations of symbolic regression for high-dimensional problems are acknowledged, and the potential for incorporating more general algorithms is discussed.

The speaker concludes by expressing excitement about the direction of polymathic AI and its potential to revolutionize scientific discovery and understanding.

Transcripts

so uh I'm very excited today to talk to

you about uh this idea of kind of

interpreting neural networks to get uh

physical Insight which I view as as kind

of a new really kind of a new paradigm

of of doing science um so this is a this

is a work with huge number of people um

I can't individually mention them all

but um many of them are here at the flat

IR Institute so I'm going to split this

up I'm going to do two parts the first

one I'm going to talk about kind of how

we go from a neural network to insights

how we actually get insights out of a

neural network the second part I'm going

to talk about this polymathic AI thing

um which is about basically building

massive uh neural networks for

science so

my motivation for this line of work is

uh examples like the

following so there was this paper led by

Kimberly stachenfeld at Deep Mind uh a

few a couple years ago on learning fast

subgrid models for fluid

turbulence um so what you see here is

the ground truth so this is kind of some

some box of a fluid uh the bottom row is

the the the Learned kind of subgrid

model essentially for this this

simulation um the really interesting

thing aart about this is that this

model was only trained on 16 simulations

but it it actually learned to be more

accurate than all traditional subgrid

models at that resolution um for fluid

dynamics so I think I think it's really

exciting kind of to figure out how did

the model do that and and kind of what

can we learn about science from this

from this uh neural

network uh another example is so this is

a work that uh I worked on with Dan too

and others on predicting instability in

planetary systems so this is a this is a

centuries old problem you have some you

know this this compact planetary system

and you want to figure out when does it

go un stable um there are literally I

mean people have literally worked on

this for

centuries um it's a fundamental problem

in chaos but this this neural network uh

trained on I think it was maybe 20,000

simulations um it's it's not only more

accurate at predicting instability but

it also seems to generalize better to

kind of different types of systems um so

it's it's really interesting to think

about okay this these neural networks

they've um they've seemed to have

learned something new how can we we

actually use that to advance our own

understanding so that's that's my

motivation here so the traditional

approach to science has been kind of you

have some low dimensional data set or

some kind of summary statistic and you

build theories to describe that uh

low-dimensional data um which might be

kind of a summary

statistic so you can look throughout the

history of science so maybe Kepler's Law

is an empirical fit to data

and then of course Newton's law of

gravitation was required to explain this

and another examples like Plank's law so

this was an actually an empirical fit to

data um and quantum mechanics was

required uh partially motivated by this

to um explain it

so this is this is uh kind of the the um

the normal approach to building theories

um and of course some of these they

they've kind of I mean it's not only

this it also involves you know many

other things but um I I think it's

really exciting to think about how we

can

involve

interpretation of datadriven models in

this process going to vary generally so

that's what I'm going to talk about

today uh I'm going to

conjecture that in this era of AI where

we have these massive neural networks

that kind of seem to outperform all of

our traditional the the um we might want

to consider this approach where we use a

neural network as essentially

compression

tool or some kind of uh tool that that

pulls apart common patterns um in uh a

data set and we build theories not to

describe the data directly but really

kind of to describe the neural network

and what the neural network has learned

um so I think this is kind of a exciting

new approach to I mean really really

science in general I think especially

the physical

sciences so the the key Point here is

neural networks trained on massive

amounts of data with with very

flexible functions they they seem to

find new things that are not in our

existing Theory so I showed you the

example with turbulence you know we can

find better subgrid models just from

data um and we can also do this with the

planetary

Dynamics so

I think our challenge as scientists for

those

problems is distilling those insights

into our language kind of incorporating

it in our Theory I think this is this is

a a really exciting way to kind of look

at these these

models so I'm going to break this down a

bit the first thing I would like to do

is just go through kind of what what

machine learning is how it works um and

then talk about this this uh kind of how

you app apply them to different data

sets Okay so just going back to the very

fundamentals uh linear regression in

1D this is I would argue if you don't

really have physical meaning to these

parameters yet it is a kind of type of

machine learning um and so this is a

it's these are scalers right X and Y

those are scalers 0 51 scalar parameters

linear

model you go One Step Beyond that and

you get this shallow Network so again

this has 1D input X 1D output y but now

we've

introduced this layer so we we have

these linear

models so we have

three hidden neurons here and they pass

through this function a so this is

called an activation function and what

this does is it gives the model a way of

uh including some

nonlinearity

so these are called activation functions

the the the one that most people would

reach for first is the rectified linear

unit or reu essentially what this does

is it says if the input is less than

zero drop it at zero greater than zero

leave it um this is a very simple way of

adding some kind of nonlinearity to my

flexible curve that I'm going to fit to

my data

right

um the next thing I do is I have these I

have these

different activation functions they have

this this kind of joint here at

different different points which depends

on the

parameters and I'm going to multiply the

output of these activations by number so

that's that's kind of the the output of

my kind of a layer of the neural network

um and this is going to maybe change the

direction of it um change the slope of

it the next thing I'm going to do is I'm

going to sum these up I'm going to

superimpose them and I get this is the

output of one layer in my network so

this is a shallow Network essentially

what it is it's a piecewise linear model

okay and the the joints here the parts

where it kind of switches from one

linear region to another those are

determined by the inputs to the the

first layers activations so it's it's

basically a piecewise linear model okay

it's a piecewise linear model um

and the one cool thing about it is you

can use this piecewise linear model to

approximate any 1D function to arbitrary

accuracy so if I want to model this

function with five joints I can get an

approximation like this with 10 joints

like this 20 like that and I can just

keep increasing the number of these

neurons that gives me better and better

approximations um so this is called the

universal approximation theorem so it's

it's that my uh shallow neural network

right it just has one one kind of layer

of activations I can describe any

continuous function um to arbitrary

Precision now that's not I mean this

alone is not uh that exciting because

like I can do that with pols right like

I don't I don't need like the neural

network is not the only thing that does

that I think the exciting part about

neural networks is when you start making

them deeper so first let's look at what

if we had two inputs what would it look

like if we had two inputs now these

activations they are activated along

planes not not points they're activated

along planes so for this is my maybe my

input plane I'm basically chopping it

along the the Zero part and now I have

these 2D planes in

space okay and the next thing I'm going

to do I'm going to scale

these and then I'm going to superimpose

them and this gives me ways of

representing

kind of arbitrary functions in now a 2d

space rather than just a 1D space so it

gives me a way of

expressing um you know arbitrary

continuous functions okay now the cool

part oops the cool part here is when I

want to do two two layers okay so now I

have two layers so I have this this is

my first neural Network this is my

second neural network and my first

neural network looks like this okay if I

consider it alone it looks like this my

second um neural network it looks like

this if I just like I cut this neural

network out it looks like this okay when

I compose them

together I get this this this shared um

kind of behavior where so I'm I'm

composing these functions together and

essentially what happens

is it's almost like you

fold the functions together so that I

experience that function in this linear

region and kind of backwards and then

again so you can see there's there's

kind of like that function is mirrored

here right it goes goes back and forth

um so you can make this analogy to

folding a piece of paper so if I

consider my first neural network like

like this on a piece of paper I could

essentially Fold It draw my second

neural network the function over that

that first one and then expand it and

essentially now I have this this uh

function

so the the cool part about this is that

I'm sharing I'm kind of sharing

computation because I'm sharing neurons

in my neural network um so this is going

to come up again this is kind of a theme

we're we're doing efficient computation

in neural networks by sharing

neurons and it's it's useful to think

about it in this this this way kind of

folding paper drawing curves over it and

expanding

it um okay so let's go back to the

physics now neural

networks uh right they're efficient

Universal function approximators you can

think of them as kind of like a type of

data

compression the same neurons can be used

for different

calculations uh in the same network um

and a common use case uh in in physical

sciences especially what I work on is

emulating physical processes so if I

have some my my simulator is kind of too

expensive or I have like real world data

my simulator is not good at describing

it I can build a neur neural network

that maybe emulates it so like I have a

neural network that looks at kind of the

initial conditions in this model and it

predicts when it's going to go

unstable so this is a this is a good use

case for them um and once I have that so

maybe I have this I have this trained

piecewise linear model that kind of

emulates some physical

process now how do I take that and go to

uh interpret it how do I actually get

insight out of

it so this is where I'm going to talk

about symbolic regression so this is one

of my favorite things so a lot of the

interpretability work in uh industry

especially like computer vision language

there's not really like there's not a

good modeling language like if I have a

if I have a model that classifies cats

and dogs there's not really like there's

not a language for

describing every possible cat there's

not like a mathematical framework for

that but in science we do have that we

do have um

oops we do have a very

good

uh mathematical

framework let me see if this

works uh so in science right so we have

this you know in science we have this

very good understanding of the

universe and

um we have this language for it we have

mathematics which describes the universe

very well uh and I think when we want to

interpret these datadriven models we

should use this language because that

will give us results that are

interpretable if I have some piece-wise

linear model with different you know

like millions of parameters it's not

it's not really useful for me right I

want to I want to express it in the

language that I'm familiar with which is

uh

mathematics um so you can look at like

any cheat sheet and it's uh it's a lot

of you know simple algebra this is the

language of

science so symbolic regression is a

machine learning task where the

objective is to find analytic

Expressions that optimize some objective

so maybe I uh maybe I want to fit that

dat set and uh what I could do is

basically try different trees so these

are like expression

trees right so this equation is that

tree and I basically find different

expression trees that uh match that data

so the point of symbolic regression I

want to find equations that fit the data

set so the symbolic and the parameters

rather than just optimizing parameters

in some

model so the the the current way to do

this the the state-of-the-art way is a

genetic algorithm so it's it's kind of

um it's not really like a clever

algorithm it's it's uh I can say that

because I work on it it's a it's it's

pretty close to Brute Force essentially

what you do is you treat your equation

like a DNA sequence and you basically

evolve it so you do like mutations you

swap one operator to another maybe maybe

you crossbreed them so you have like two

expressions which are okay you literally

breed those together I mean not

literally but you conceptually breed

those together get a new expression um

until you fit the data set

um

so yeah so this is a genetic algorithm

based search uh for symbolic regression

now

the the point of this is uh to find

simple models in our language of

mathematics that describe uh a given

data

set so um so I've spent a lot of time

working on these Frameworks so piser

symbolic regression.

JL um

they they work like this so if I have

this expression I want to model that

data set essentially what I'm going to

do is just search over all possible

Expressions uh until I find one that

gets me closer to this ground truth

expression so you see it's kind of

testing different different branches in

evolutionary space I'm going to play

that

again until it reaches this uh ground

truth data set so this is this is pretty

close to how it

works uh you're essentially finding

simple Expressions that fit some data

set accurately

okay

so what I'm going to show you how to do

is this symbolic regression idea is

about fitting kind of finding models

symbolic models that I can use to

describe a data set I want to use that

to build surrogate models of my neural

network so this is this is kind of a way

of translating my model into my language

you could you could also think of it as

like polom uh or like a tailor expansion

in some

ways the way this works is as

follows if I have some neural network

that I've trained on my data set

whatever I'm going to train it normally

freeze the

parameters then what I do is

I record the inputs and outputs I kind

of treat it like a data generating

process I I try to see like okay what's

the behavior for this input this input

and so on then I stick those inputs and

outputs into piser for example and I I

find some equation that models that

neural network or maybe it's like a

piece of my neural

network so this is a this is building a

surrogate model for my neural network

that is kind of a a Pro imates the same

behavior now you wouldn't just do this

for like a standalone neural network

this this would typically be part of

like a larger model um and it would give

you a way of interpreting exactly what

it's doing for different

inputs so what I might have is maybe I

have like two two pieces like two neural

networks here maybe I think the first

neural network is like learning features

or it's learning some kind of coordinate

transform the second one is doing

something in that space uh it's using

those features for

calculation um and so I can using

symbolic regression uh which we call

symbolic distillation I can I can

distill this model uh into

equations so that's that's the basic

idea of this I

replace neural networks so I replaced

them with my surate model which is now

an equation

um you would typically do this for G as

well and now I have equations that

describe my

model um and this is kind of a a

interpretable approximation of my

original neural network now the reason

you wouldn't want to do this for like

just directly on the data is because

it's a harder search problem if you

break it into

pieces like kind of interpreting pieces

of a neural network it's easier because

you're only searching for

2 N Expressions rather than n s so it's

a it's a bit easier and you're kind of

using the Neal Network as a way of

factoring factorizing the system into

different pieces that you then

interpret um so we've we've used this in

in different papers so this is one uh

led

by Pablo Lemos on uh rediscovering

Newton's law of gravity from data

so this was a this was a cool paper

because we didn't tell it the masses of

the bodies in the solar system it had to

simultaneously find the masses of every

all of these 30 bodies we gave it and it

also found the law um so we kind of

train this neural network to do this and

then we interpret that neural network

and it gives us uh Newton's law of

gravity um now that's a rediscovery and

of course like we know that so I think

the discoveries are also cool so these

are not my papers these are other

people's papers I thought they were

really exciting so this is one a recent

one by Ben Davis and jial Jinn where

they discover this new uh blackhole Mass

scaling

relationship uh so it's uh it relates

the I think it's the spirality or

something in a galaxy in the velocity

with the mass of a black hole um so they

they found this with this technique uh

which is exciting um and I saw this

other cool one recently um they found

this cloud cover model with this

technique uh using piser um so they it

kind of gets you this point where it's a

it's a fairly simple model and it's also

pretty accurate um but again the the

point of this is to find a model that

you can understand right it's not this

blackbox neural network with with

billions of parameters it's a it's a

simple model that you can have a handle

on okay so that's part one now part two

I want to talk about polymathic AI so

this is kind of like the complete

opposite end we're going to go from

small models in the first part now we're

going to do the biggest possible models

um and I'm going to also talk about the

meaning of Simplicity what it actually

means so

the past few years you may have noticed

there's been this shift in indust

industrial machine learning to favor uh

Foundation models so like chat GPT is an

example of this a foundation model is a

machine learning model that serves as

the foundation for other

models these models are trained by

basically taking massive amounts of

General diverse data

uh and and training this flexible model

on that data and then fine-tuning them

to some specific task so you could think

of it as maybe teaching this machine

learning model English and French before

teaching it to do translation between

the two um so it often gives you better

performance on Downstream tasks I mean

you can also see that I mean Chad gbt is

uh I've heard that it's trained on um

GitHub and that kind of teaches it to uh

reason a bit better um and so the I mean

basically these models are trained on

massive amounts of data um and they form

this idea called a foundation

model so um the general idea is you you

collect you know you collect your

massive amounts of data you have this

very Flex ible model and then you train

it on uh you might train it to do uh

self supervised learning which is kind

of like you mask parts of the data and

then the model tries to fill it back in

uh that's a that's a common way you

train that so like for example GPT style

models those are basically trained on

the entire internet and they're trained

to predict the next word that's that's

their only task you get a input sequence

of words you predict the next one and

you just repeat that for uh massive

amounts of text and then just by doing

that they get really good at um General

language understanding then they are

fine-tuned to be a chatbot essentially

so they're they're given a little bit of

extra data on uh this is how you talk to

someone and be friendly and so on um and

and that's much better than just

training a model just to do that so it's

this idea of pre-training

models so I mean once you have this

model I I think like kind of the the the

cool part about these models is they're

really trained in a way that gives them

General priors for data so if I have

like some maybe I have like some artwork

generation model it's trained on

different images and it kind of

generates different art

I can fine-tune this model on like

studio gibli artartwork and it doesn't

need much training data because it

already knows uh what a face looks like

like it's already seen tons of different

faces so just by fine tuning it on some

small number of examples it can it can

kind of pick up this task much quicker

that's that's essentially the idea

now this is I mean the same thing is

true in language right like if I if I

train a model on uh if I train a model

just to do language

translation right like I just teach it

that it's kind of I start from scratch

and I just train it English to French um

it's going to struggle whereas if I

teach it English and French kind of I I

teach it about the languages first and

then I specialize it on translation um

it's going to do much

better so this brings us to science so

in

um in science we also have this we also

have this idea where there are shared

Concepts right like different languages

have shared there's shared concept of

grammar in different languages in

science we also have shared Concepts you

could kind of draw a big circle around

many areas of Science and causality is a

shared concept uh if you zoom in to say

dynamical systems um you could think

about like multiscale Dynamics is is

shared in many different disciplines uh

chaos is another shared concept

so maybe if we train a general

model uh you know over many many

different data sets the same way Chad

GPT is trained on many many different

languages and and text databases maybe

they'll pick up general concepts and

then when we finally make it specialize

to our particular problem uh maybe

they'll do it it'll find it easier to

learn so that's essentially the

idea so you can you can really actually

see this for particular systems so one

example is the reaction diffusion uh

equation this is a type of PD um and the

shallow water equations another type of

PD different fields different pdes but

both have

waves so they they both have wav like

Behavior so I mean maybe if we train

this massive flexible model on both of

these system it's going to kind of learn

a general prior for uh what a wave looks

like and then if I have like some you

know some small data set I only have a

couple examples of uh maybe it'll

immediately identify oh that's a wave I

know how to do that um it's it's almost

like I mean I kind of feel like in

science today what we often do

is I mean we train machine learning

models from scratch it's almost like

we're taking uh Toddlers and we're

teaching them to do pattern matching on

like really Advanced problems like we we

have a toddler and we're showing them

this is a you know this is a spiral

galaxy this is an elliptical galaxy and

it it kind of has to just do pattern

matching um whereas maybe a foundation

model that's trained on broad classes of

problems um it's it's kind of like a

general uh science graduate maybe um so

it has a prior for how the world works

it has seen many different phenomena

before and so when it when you finally

give it that data set to kind of pick up

it's already seen a lot of that

phenomena that's that's really the of

this uh that's why we think this will

work

well okay so we we created this

collaboration last year uh so this

started at flat iron Institute um led by

Shirley ho to

build this thing a foundation model for

science so this uh this is across

disciplines so we want to you know build

these models to incorporate data across

many different disciplines uh across

institutions um and uh so we're we're

currently working on kind of scaling up

these models right now the

final I think the final goal of this

collaboration is that we would release

these open-source Foundation models so

that people could download them and and

fine-tune them to different tasks so

it's really kind of like a different

Paradigm of doing machine learning right

like rather than the current Paradigm

where we take a model randomly

initialize it it's kind of like a like a

toddler doesn't know how the world Works

um and we train that this Paradigm is we

have this generalist science model and

you start from that it's kind of a

better initialization of a

model that's that's the that's the pitch

of

polymathic okay so we have results so

this year we're kind of scaling up but

uh last year we had a couple papers so

this is one uh led by Mike mccab called

multiple physics

pre-training this paper looked at what

if we have this General PD simulator

this this model that learns to

essentially run fluid Dynamic

simulations and we train it on many

different PDS will it do better on new

PDS or will it do worse

uh so what we found is that a single so

a single model is not only able to match

uh you know single uh single models

trained on like specific tasks it can

actually outperform them in many cases

so it it does seem like if you take a

more flexible model you train it on more

diverse

data uh it will do better in a lot of

cases I mean it's it's not

unexpected um because we do see this

with language and vision um but I I

think it's still really cool to uh to

see

this so um I'll skip through some of

these so this is like this is the ground

truth data and this is the

Reconstruction essentially what it's

doing is it's predicting the next step

all right it's predicting the next

velocity the next density and pressure

and so on and you're taking that

prediction and running it back through

the model and you get this this roll out

simulation so this is a this is a task

people work on in machine

learning um I'm going to skip through

these uh and essentially what we found

is that uh most of the time by uh using

this multiple physics pre-training so by

training on many different PDS you do

get better performance so the ones at

the right side are the uh multiple

physics pre-trained models those seem to

do better in many cases and it's really

because I mean I think because they've

seen you know so many different uh PDS

it's like they have a better prior for

physics

um skip this as well so okay this is a

funny thing that we observed is that

so during talks like this one thing that

we get asked is how similar do the PDS

need to be like do the PDS need to be

you know like navor Stokes but a

different

parameterization or can they be like

completely different physical systems so

what we found

is uh

really uh hilarious is that okay so the

bottom line here this is the air of the

model

uh over different number of training

examples so this model was trained on a

bunch of different PDS and then it was

introduced to this new PD problem and

it's given that amount of data okay so

that does the best this model it's

already it already knows some Physics

that one does the best the one at the

top is the worst this is the model

that's trained from scratch it's never

seen anything uh this is like your

toddler right like it's never it doesn't

know how the physical world Works um it

was just randomly initialized and it has

to learn physics okay the middle models

those are pre-trained on General video

data a lot of which is Cap videos so

even pre-training this model on cap

videos actually helps you do much better

than this very sophis phisticated

Transformer architecture that just has

never seen any data and it's really

because I mean we think it's because of

shared concepts of spaciotemporal

continuity right like videos of cats

there's a you know there's there's a

spaciotemporal

continuity like the cat does not

teleport across the video unless it's a

very fast cat um there's related

Concepts right so I mean that's that's

what we think but it's it's really

interesting that uh you know

pre-training on completely unrelated

systems still seems to help

um and so the takeaway from this is that

you should always pre-train your model

uh even if the physical system is not

that related you still you still see

benefit of it um now obviously if you

pre-train on related data that helps you

more but anything is basically better

than than nothing you could basically

think of this as the

default initialization for neural

networks is garbage right like just

randomly initializing a neural network

that's a bad starting point it's a bad

prior for physics you should always

pre-train your model that's the takeaway

of this okay so um I want to finish up

here with kind of rhetorical questions

so I started the talk about um

interpretability and kind of like how do

we extract insights from our model now

we've we've kind of gone into this

regime of these very large very flexible

Foundation models that seem to learn

general

principles so okay my question for you

you don't have to answer but just think

it over is do you think 1 + 1 is

simple it's not a trick question do you

think 1 + 1 is simple so I think most

people would say yes 1+ 1 is

simple and if you break that down into

why it's simple you say okay so X Plus Y

is simple for like X and Y integers

that's a simple relationship okay why Y

is X Plus y

simple and and you break that down it's

because plus is simple like plus is a

simple operator okay why why is plus

simple it's a very abstract

concept okay it's it's we we don't

necessarily have plus kind of built into

our brains um it's it's kind of I mean

it's it's really

uh so I'm going to show this this might

be controversial but I think that

Simplicity is based on familiar

we are used to plus as a concept we are

used to adding numbers as a concept

therefore we call it

simple you can go back another step

further the reason we're familiar with

addition is because it's useful adding

numbers is useful for describing the

world I count things right that's useful

to live in our universe it's useful to

count things to measure things addition

is

useful and it's it's it's really one of

the most useful things so that is why we

are familiar with it and I would argue

that's why we think it's

simple but the the Simplicity we have

often argued is uh if it's simple it's

more likely to be useful I think that is

actually not a statement about

Simplicity it's actually a statement

that if if something is useful for

problems like a b and c then it seems it

will also be useful for another problem

the the the world is compositional if I

have a model that works for this set of

problems it's probably also going to

work for this one um so that's that's

the argument I would like to make so

when we interpret these models I think

it's important to kind of keep this in

mind and and and really kind of probe

what is simple what is

interpretable

so I think this is really exciting for

polymathic AI because these models that

are trained on many many systems they

will find broadly useful algorithms

right they'll they'll they'll have these

neurons that share calculations across

many different disciplines so you could

argue that that is the utility and I

mean like maybe we'll discover new kind

of operators and be familiar with those

and and and we'll start calling those

simple so it's not necessarily that all

of the uh things we discover in machine

learning will be uh simple it it's uh

kind of that by definition the polymath

models will be broadly

useful and if we know they're broadly

useful we might we might might get

familiar with those and and that might

kind of Drive the Simplicity of them um

so that's my node on Simplicity and so

the the takeaways here are that I think

interpreting a neural

network trained on some data sets um

offers new ways of discovering

scientific insights from that data um

and I I think Foundation models like

polyic AI I think that is a very

exciting way of discovering new broadly

applicable uh scientific models so I'm

really excited about this direction uh

and uh thank you for listening to me

[Applause]

today great U so three

questions one was the

running

yeah when it's fully built out is to be

free

yeah please use your seat

mic

yeah and three

you're pretty

young okay so I'll try to

compartmentalize those okay so the first

question was the scale of training um

this is really an open research question

we don't have the scaling law for

science yet we have scaling laws for

language we know that if you have this

many gpus you have this size data set

this is going to be your performance we

don't have that yet for science cuz

nobody's built this scale of model um so

that's something we're looking at right

now is what is the tradeoff of scale and

if I want to train this model on many

many gpus is it is it worth it um so

that's an that's an open research

question um I do think it'll be large

you know

probably order hundreds of gpus uh

trained for um um maybe a couple months

um so it's going to be a very large

model um that's that's kind of assuming

the scale of language models um now the

model is going to be free definitely

we're we're uh we're all very Pro open

source um and I think that's I mean I

think that's really like the point is we

want to open source this model so people

can download it and use it in science I

think that's really the the most

exciting part about this um and then I

guess the Third question you had was

about the future um and how it

changes uh how we

teach um I mean I guess uh are you are

you asking about teaching science or

teaching machine learning teaching

science I see

um I mean yeah I mean I don't know it

depends if it if it works I think if it

works it it might very well like change

how how science is taught

um yeah I mean so I don't I don't know

the impact of um language models on

computational Linguistics I'm assuming

they've had a big impact I don't know if

that's affected the teaching of it yet

um but if if you know scientific

Foundation models had a similar impact

I'm sure I'm sure it would impact um I

don't know how much it probably depends

on the success of the

models I I have a question about your

foundation models also so in different

branches of science the data sets are

pretty different in molecular biology or

genetics the data sets you know is a

sequence of DNA versus astrophysics

where it's images of stars so how do you

plan to you know use the same model you

know for different different form of

data sets input data sets uh so you mean

how to pose the objective yes so I I

think the most I mean the most General

objective is self-supervised learning

where you basically mask parts of the

data and you predict the missing part if

you can you know optimize that problem

then you can solve tons of different

ones you can do uh regression predict

parameters or go the other way and

predict rollouts of the model um it's a

really General problem to mask data and

then fill it back in that kind of is a

superset of uh many different prediction

problems yeah and I think that's why

like language models are so broadly

useful even though there train just on

next word prediction or like B is a

masked

model thanks uh can you hear me all

right so um that was a great talk um I'm

Victor uh so uh I'm actually a little

bit uh worried and this is a little bit

of a question whenever you have models

like this um you said that you train

this on many examples right so imagine

you have already embedded the laws of

physics here somehow like let's say the

law of ration but when you when you

think about like this c new physics we

always have this question whether we are

you know actually Reinventing the wheel

or like the uh the network is kind of

really giving us something new or is it

something giving us uh or it's giving us

something that you know it it learned

but it's kind of wrong so in sometimes

we have the answer to know you know

which one is which but if you don't have

that let's say for instance you're

trying to discover what dark matter is

which you know something I'm working on

how would you know that the networ is

actually giving you something new and

not you know just trying to set this

into one of the many parameters that it

has I see um

so okay

so so if you want to test the model by

letting it ReDiscover something then I

don't think you should use this I think

you should use the scratch model like

from scratch and train it because if you

TR if you use a pre-train model it's

probably already seen that physics so

it's biased towards it in some ways so

if you're rediscovering something I

don't think you should use this if

you're discovering something new um I do

think this is more useful um so I think

a like a a

misconception of of uh I think machine

learning in general is that scientists

view machine learning for uninitialized

models like randomly initialized weights

as a neutral prior but it's not it's a

very uh it's a very explicit prior um

and it happens to be a bad prior um so

if you train from a a randomly

initialized model it's it's kind of

always going to be a worse prior than

training from a pre-train model which

has seen many different types of physics

um I think I think we can kind of make

that statement um so if you're if you're

trying to discover new physics I I mean

I mean like if it if you train it on

some data set um I guess you can always

verify that it that the predictions are

accurate so that would be um I guess one

way to to verify it um but I I do think

like the fine-tuning here so like taking

this model and training it on the task I

think that's very important I think in

language models it's not it's not as

emphasized like people will just take a

language model and and tweak the prompt

to get a better result I think for

science I think the prompt is I mean I

think like the equivalent of the prompt

would be important but I think the fine

tuning is much more important because

our data sets are so much different

across

science the

back that the

symbolic lied the dimensionality of the

system so are you introducing also the

funing and transfer learning a

way

en uh yeah

so so the symbolic regression I mean I

would consider that it it's not used

inside the foundation model part I think

it's

interesting to interpret the foundation

model and see if there's kind of more

General physical Frameworks that it

comes up with

um I think yeah symbolic regression is

very limited in that it's bad at high

dimensional

problems I think that might

be because of the choice of operators um

like I think if you can consider maybe

High dimensional operators you you might

be uh a bit better off I mean symbolic

regression it it's uh it's an active

area of research and I think the hardest

the biggest hurdle right now is it's uh

it's not good at finding very complex

symbolic

models

comp so um I guess uh you

could it depends like on the

dimensionality of the data

um I guess if it's very high dimensional

data you're always kind of um like

symbolic regression is not good to high

dimensional data unless you can have

kind of some operators that aggregate to

lower dimensional uh

spaces um I don't yeah I don't know if

I'm answering your question or

not okay I wanted to ask a little bit so

like when you were showing the

construction of these trees each

generation in the different operators I

think this is related to kind of General

themes of the talk and other questions

but often in doing science when you're

learning it you're presented with kind

of like algi to solve problems like you

know diagonalize hilon or something like

that what how do you encapsulate that

aspect of doing science that is kind of

the almic side soling problem

rather right please use your mic oh yeah

uh yeah so the question was about um how

do you incorporate kind of more General

uh not analytic operators but kind of

more General algorithms like a

hamiltonian operator um I think that I

mean like in principle symbolic

regression is it's part of a larger

family of an algorithm called program

synthesis where the objective is to find

a program you know like code that

describes a given data set for example

so

if you can write your

operators into your symbolic regression

approach and your symbolic regression

approach has that ground truth model in

there somewhere then I think it's

totally possible I think like it's it's

uh it's harder to do I think like even

symbolic regression with scalers is uh

it's fairly it's fairly difficult to to

actually set up an algorithm um I think

I don't know I think it's really like an

engineering problem but the the the

conceptual part is uh is totally like

there for this

yeah thanks um oh

sorry okay um this this claim uh that

random initial weights are always bad or

pre-training is always good I don't know

if they're always bad but um it seems

like from our

experiments it's we've never seen a case

where

pre-training um on some kind of physical

data hurts like the cap video is is an

example we thought that would hurt the

model it didn't that is a cute example

weird I'm sure there's cases where some

pre-training hurts yeah so that that's

essentially my question so we're aware

of like adversarial examples for example

you train on Mist add a bit of noise it

does terrible compared to what a human

buo what do you think adversarial

examples look like in science yeah yeah

I mean I don't I don't know what those

are but I'm sure they exist somewhere

where pre-training on certain data types

kind of messes with training a bit um we

don't know those yet but uh yeah it'll

be interesting do you think it's a

pitfall though of like the approach

because like I have a model of the sun

and a model of DNA you know it's yeah

yeah I mean um I don't know like um I

guess we'll see um yeah it's it's hard

to it's hard to know like I guess from

language we've seen you can pre-train

like a language model on video data and

it helps the language which is really

weird but it it does seem like if

there's any kind of Concepts it does if

it's flexible enough it can kind of

transfer those in some ways so we'll see

I mean there's I mean presumably we'll

find some adversarial examples there so

far we haven't we thought the cat was

one but it wasn't it it

helped