Overview
============================

The name of this book, _Physics-Based Deep Learning_,
denotes combinations 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. The following chapter will
give a more thorough introduction to the topic and establish the basics
for following chapters.

```{figure} resources/overview-pano.jpg
---
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 next-gen simulations
could strongly profit from integrating deep learning components to make even 
more accurate predictions about our world.
```

## Motivation

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
become an integral part of science.  

In recent years, machine learning technologies and _deep neural networks_ in particular,
have led to impressive achievements in a variety of fields:
from image classification {cite}`krizhevsky2012` over
natural language processing {cite}`radford2019language`, 
and more recently also for protein folding {cite}`alquraishi2019alphafold`.
The field is very vibrant and quickly developing, with the promise of vast possibilities.

These success stories of deep learning (DL) approaches 
have given rise to concerns that this technology has 
the potential to replace the traditional, simulation-driven approach to science. 
E.g., recent works show that NN-based surrogate models achieve accuracies required
for real-world, industrial applications such as airfoil flows {cite}`chen2021highacc`, while at the
same time outperforming traditional solvers by orders of magnitude in terms of runtime.

Instead of relying on models that are carefully crafted
from first principles, can data collections of sufficient size
be processed to provide the correct answers?
As we'll show in the next chapters, this concern is unfounded. 
Rather, it is crucial for the next generation of simulation systems
to bridge both worlds: to 
combine _classical numerical_ techniques with _deep learning_ methods.

One central reason for the importance of this combination is
that DL approaches are powerful, but at the same time strongly profit
from domain knowledge in the form of physical models.
DL techniques and NNs are novel, sometimes difficult to apply, and
it is admittedly often non-trivial to properly integrate our understanding
of physical processes into the learning algorithms.

Over the course of the last decades,
highly specialized and accurate discretization schemes have
been developed to solve fundamental model equations such
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
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
The key aspects that we want to address in the following are:
- explain how to use deep learning techniques to solve PDE problems,
- how to combine them with **existing knowledge** of physics,
- without **discarding** our knowledge about numerical methods.
```

Thus, our aim is to build on all the powerful techniques that we have
at our disposal, and use them wherever we can.
As such, a central goal of this book is to _reconcile_ the data-centered
viewpoint with physical simulations.

The resulting methods have a huge potential to improve
what can be done with numerical methods: in scenarios
where a solver targets cases from a certain well-defined problem
domain repeatedly, it can for instance make a lot of sense to once invest 
significant resources to train
a neural network that supports the repeated solves. Based on the
domain-specific specialization of this network, such a hybrid
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 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
observations).

![An overview of categories of physics-based deep learning methods](resources/physics-based-deep-learning-overview.jpg)

No matter whether we're considering forward or inverse problems, 
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
via partial differential equations (PDEs).
Taking a global perspective, the following three categories can be
identified to categorize _physics-based deep learning_ (PBDL)
techniques:

- _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. 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 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 
the interleaved approaches
that leverage _differentiable physics_ allow for very tight integration
of deep learning and numerical simulation methods.


## More specifically

_Physical simulations_ are a huge field, and we won't cover all possible types of physical models and simulations in the following.

```{note} Rather, the focus of this book lies on:
- _Field-based simulations_ (no Lagrangian methods)
- Combinations with _deep learning_ (plenty of other interesting ML techniques, but not here)
- Experiments as _outlook_ (i.e., replace synthetic data with real-world observations)
```

It's also worth noting that we're starting to build the methods from some very
fundamental building blocks. Here are some considerations for skipping ahead to the later chapters.

```{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 {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 {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!
```

## Implementations

This text also represents an introduction to a wide range of deep learning and simulation APIs.
We'll use popular deep learning APIs such as _pytorch_ [https://pytorch.org](https://pytorch.org) and _tensorflow_ [https://www.tensorflow.org](https://www.tensorflow.org), and additionally
give introductions into the differentiable simulation framework _Φ<sub>Flow</sub> (phiflow)_ [https://github.com/tum-pbs/PhiFlow](https://github.com/tum-pbs/PhiFlow). Some examples also use _JAX_ [https://github.com/google/jax](https://github.com/google/jax). Thus after going through
these examples, you should have a good overview of what's available in current APIs, such that
the best one can be selected for new tasks.

As we're (in most jupyter notebook examples) dealing with stochastic optimizations, many of the following code examples will produce slightly different results each time they're run. This is fairly common with NN training, but it's important to keep in mind when executing the code. It also means that the numbers discussed in the text might not exactly match the numbers you'll see after re-running the examples.

---
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<!-- ## A brief history of PBDL in the context of Fluids

First:

Tompson, seminal...

Chu, descriptors, early but not used

Ling et al. isotropic turb, small FC, unused?

PINNs ... and more ... -->

