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Table of Contents
I Introduction
1 A Teaser Example
1.1 Differentiable physics
1.2 Finding the inverse function of a parabola
1.3 A differentiable physics approach
1.4 Discussion
1.5 Next steps
2 Overview
2.1 Motivation
2.2 Categorization
2.3 Looking ahead
2.4 Implementations
2.5 Models and Equations
2.6 Simple Forward Simulation of Burgers Equation with phi ow
2.7 Navier-Stokes Forward Simulation
3 Supervised Training
3.1 Problem setting
3.2 Surrogate models
3.3 Show me some code!
3.4 Supervised training for RANS ows around airfoils
3.5 Discussion of Supervised Approaches
II Physical Losses
4 Physical Loss Terms
4.1 Using physical models
4.2 Variant 1: Residual derivatives for explicit representations
4.3 Variant 2: Derivatives from a neural network representation
4.4 Summary so far
5 Burgers Optimization with a Physics-Informed NN
5.1 Formulation
5.2 Preliminaries
5.3 Loss function and training
5.4 Evaluation
5.5 Next steps
6 Discussion of Physical Losses
6.1 Is it Machine Learning?
6.2 Summary
III Differentiable Physics
7 Introduction to Differentiable Physics
7.1 Differentiable operators
7.2 Jacobians
7.3 Learning via DP operators
7.4 A practical example
7.5 Implicit gradient calculations
7.6 Summary of differentiable physics so far
8 Burgers Optimization with a Differentiable Physics Gradient
8.1 Initialization
8.2 Gradients
8.3 Optimization
8.4 More optimization steps
8.5 Physics-informed vs. differentiable physics reconstruction
8.6 Next steps
9 So Far so Good - a First Discussion
9.1 Compatibility with existing numerical methods
9.2 Discretization
9.3 Efficiency
9.4 Efficiency continued
9.5 Summary
10 Differentiable Fluid Simulations
10.1 Physical Model
10.2 Formulation
10.3 Starting the Implementation
10.4 Batched simulations
10.5 Gradients
10.6 Optimization
10.7 Re-simulation
10.8 Conclusions
10.9 Next steps
IV Differentiable Physics with NNs
11 Integrating DP into NN Training
11.1 Alternatives: noise
11.2 Complex Examples
12 Reducing Numerical Errors with Deep Learning
12.1 Problem formulation
12.2 Getting started with the implementation
12.3 Simulation setup
12.4 Network architecture
12.5 Data handling
12.6 Interleaving simulation and NN
12.7 Training
12.8 Evaluation
12.9 Next steps
Summary
This document discusses the intersection of physics-based methodologies and deep learning techniques. It covers topics such as differentiable physics, supervised training, physical loss terms, optimization methods, and fluid simulations. The document provides insights into integrating differentiable physics into neural network training and reducing numerical errors using deep learning. It also explores various examples and practical applications within the field.