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18
intro.md
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@ -10,6 +10,8 @@ As much as possible, the algorithms will come with hands-on code examples to qui
Beyond standard _supervised_ learning from data, we'll look at _physical loss_ constraints,
more tightly coupled learning algorithms with _differentiable simulations_, as well as extensions such
as reinforcement learning and uncertainty modeling.
These methods have a huge potential to fundamentally change what we can achieve
with simulations.
```{figure} resources/teaser.jpg
@ -30,6 +32,14 @@ As a _sneak preview_, in the next chapters will show:
- How to more tightly interact with a full simulator for _inverse problems_. E.g., we'll demonstrate how to circumvent the convergence problems of standard reinforcement learning techniques by leveraging simulators in the training loop.
The different PBDL techniques will be introduced ordered in terms of growing
tightness of the integration, and pros and cons of the different approaches
will be discussed. It's important to know in which scenarios each of the
different techniques is particularly useful.
## Comments and suggestions
This _book_, where "book" stands for a collection of texts, equations, images and code examples,
is maintained by the
[TUM Physics-based Simulation Group](https://ge.in.tum.de). Feel free to contact us
@ -55,22 +65,24 @@ Plus, jupyter notebooks are great because they're a form of [literate programmin
This project would not have been possible without the help of many people who contributed. Thanks to everyone 🙏 Here's an alphabetical list:
- [Philipp Holl](https://ge.in.tum.de/about/)
% - [Georg Kohl](https://ge.in.tum.de/about/georg-kohl/)
- [Maximilian Mueller](https://ge.in.tum.de/)
- [Patrick Schnell](https://ge.in.tum.de/about/patrick-schnell/)
- [Felix Trost](https://ge.in.tum.de/)
- [Nils Thuerey](https://ge.in.tum.de/about/n-thuerey/)
- [Kiwon Um](https://ge.in.tum.de/about/kiwon/)
% - [Georg Kohl](https://ge.in.tum.de/about/georg-kohl/)
## Citation
If you find this book useful, please cite via:
If you find this book useful, please cite it via:
```
@article{thuerey2021pbdl,
title={Physics-based Deep Learning},
author={Thuerey, Nils and Maximilian Mueller and Patrick Schnell and Felix Trost and Kiwon Um},
author={Nils Thuerey and Maximilian Mueller and Patrick Schnell and Felix Trost and Kiwon Um},
url={http://physicsbaseddeeplearning.org},
year={2021},
publisher={www}
}
```

57
overview.md
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@ -1,17 +1,13 @@
Overview
============================
The following collection of digital documents, i.e. "book",
targets _Physics-Based Deep Learning_ techniques.
By that we mean combining physical modeling and numerical simulations with
The name of this book, _Physics-Based Deep Learning_,
denotes combiniations of physical modeling and numerical simulations with
methods based on artificial neural networks.
The general direction of Physics-Based Deep Learning represents a very
active, quickly growing and exciting field of research -- we want to provide
a starting point for new researchers as well as a hands-on introduction into
state-of-the-art research topics.
active, quickly growing and exciting field of research, and the following chapter will
give a more thorough introduction for the topic and establish the basics
for following chapters.
```{figure} resources/overview-pano.jpg
---
@ -19,11 +15,13 @@ height: 240px
name: overview-pano
---
Understanding our environment, and predicting how it will evolve is one of the key challenges of humankind.
A key tool for achieving these goals are simulations, and the next generation of simulation algorithms
will rely heavily on deep learning components to yield even more accurate predictions about our world.
```
## Motivation
From weather and climate forecasts {cite}`stocker2014climate`,
From weather and climate forecasts {cite}`stocker2014climate` (see the picture above),
over quantum physics {cite}`o2016scalable`,
to the control of plasma fusion {cite}`maingi2019fesreport`,
using numerical analysis to obtain solutions for physical models has
@ -47,7 +45,7 @@ it is crucial to bring together both worlds: _classical numerical techniques_
and _deep learning_.
One central reason for the importance of this combination is
that DL approaches are simply not powerful enough by themselves.
that DL approaches are simply not yet powerful enough by themselves.
Given the current state of the art, the clear breakthroughs of DL
in physical applications are outstanding.
The proposed techniques are novel, sometimes difficult to apply, and
@ -62,12 +60,12 @@ as the Navier-Stokes, Maxwell’s, or Schroedinger’s equations.
Seemingly trivial changes to the discretization can determine
whether key phenomena are visible in the solutions or not.
Rather than discarding the powerful methods that have been
carefully developed in the field of numerical mathematics, it
developed in the field of numerical mathematics, it
is highly beneficial for DL to use them as much as possible.
```{admonition} Goals of this document
:class: tip
Thus, the key aspects that we want to address in the following are:
The key aspects that we want to address in the following are:
- explain how to use deep learning techniques,
- how to combine them with **existing knowledge** of physics,
- without **throwing away** our knowledge about numerical methods.
@ -80,7 +78,7 @@ viewpoint and the physical simulation viewpoint.
The resulting methods have a huge potential to improve
what can be done with numerical methods: e.g., in scenarios
where solves target cases from a certain well-defined problem
where a solver targets cases from a certain well-defined problem
domain repeatedly, it can make a lot of sense to once invest
significant resources to train
an neural network that supports the repeated solves. Based on the
@ -89,12 +87,17 @@ could vastly outperform traditional, generic solvers. And despite
the many open questions, first publications have demonstrated
that this goal is not overly far away.
Another way to look at it is that all mathematical models of our nature
are idealized approximations and contain errors. A lot of effort has been
made to obtain very good model equations, but in order to make the next
big step forward, DL methods offer a very powerful tool to close the
remaining gap towards reality.
## Categorization
Within the area of _physics-based deep learning_,
we can distinguish a variety of different
approaches, from targeting designs, constraints, combined methods, and
approaches, from targeting constraints, combined methods, and
optimizations to applications. More specifically, all approaches either target
_forward_ simulations (predicting state or temporal evolution) or _inverse_
problems (e.g., obtaining a parametrization for a physical system from
@ -106,39 +109,33 @@ No matter whether we're considering forward or inverse problem,
the most crucial differentiation for the following topics lies in the
nature of the integration between DL techniques
and the domain knowledge, typically in the form of model equations.
Looking ahead, we will particularly aim for a very tight integration
of the two, that goes beyond soft-constraints in loss functions.
Taking a global perspective, the following three categories can be
identified to categorize _physics-based deep learning_ (PBDL)
techniques:
- _Data-driven_: the data is produced by a physical system (real or simulated),
but no further interaction exists.
- _Supervised_: the data is produced by a physical system (real or simulated),
but no further interaction exists. This is the classic machine learning approach.
- _Loss-terms_: the physical dynamics (or parts thereof) are encoded in the
loss function, typically in the form of differentiable operations. The
learning process can repeatedly evaluate the loss, and usually receives
gradients from a PDE-based formulation.
gradients from a PDE-based formulation. These soft-constraints sometimes also go
under the name "physics-informed" training.
- _Interleaved_: the full physical simulation is interleaved and combined with
an output from a deep neural network; this requires a fully differentiable
simulator and represents the tightest coupling between the physical system and
the learning process. Interleaved approaches are especially important for
the learning process. Interleaved differentiable physics approaches are especially important for
temporal evolutions, where they can yield an estimate of future behavior of the
dynamics.
Thus, methods can be roughly categorized in terms of forward versus inverse
solve, and how tightly the physical model is integrated into the
optimization loop that trains the deep neural network. Here, especially approaches
optimization loop that trains the deep neural network. Here, especially
the interleaved approaches
that leverage _differentiable physics_ allow for very tight integration
of deep learning and numerical simulation methods.
The goal of this document is to introduce the different PBDL techniques,
ordered in terms of growing tightness of the integration, give practical
starting points with code examples, and illustrate pros and cons of the
different approaches. In particular, it's important to know in which scenarios
each of the different techniques is particularly useful.
## More specifically
@ -156,9 +153,9 @@ fundamental steps. Here are some considerations for skipping ahead to the later
```{admonition} Hint: You can skip ahead if...
:class: tip
- you're very familiar with numerical methods and PDE solvers, and want to get started with DL topics right away. The _Supervised Learning_ chapter is a good starting point then.
- you're very familiar with numerical methods and PDE solvers, and want to get started with DL topics right away. The {doc}`supervised` chapter is a good starting point then.
- On the other hand, if you're already deep into NNs&Co, and you'd like to skip ahead to the research related topics, we recommend starting in the _Physical Loss Terms_ chapter, which lays the foundations for the next chapters.
- On the other hand, if you're already deep into NNs&Co, and you'd like to skip ahead to the research related topics, we recommend starting in the {doc}`physicalloss` chapter, which lays the foundations for the next chapters.
A brief look at our _notation_ in the {doc}`notation` chapter won't hurt in both cases, though!
```