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Welcome …

Introduction

A Teaser Example
Overview

Neural Surrogates and Operators

Supervised Training
Neural Network Architectures
Supervised training for RANS flows around airfoils
Discussion of Supervised Approaches

Physical Losses

Physical Loss Terms
Learning the Helmholtz-Hodge Decomposition
Burgers Optimization with a PINN
Discussion of Physical Losses

Differentiable Physics

Introduction to Differentiable Physics
Burgers Optimization with a Differentiable Physics Gradient
So Far so Good - a First Discussion
Differentiable Fluid Simulations
Integrating DP into NN Training
Reducing Numerical Errors with Neural Operators
Solving Inverse Problems with NNs
Discussion of Differentiable Physics

Probabilistic Learning

Introduction to Probabilistic Learning
Learning a Probability Distribution
Score Matching
Denoising
Flow Matching
Denoising and Flow Matching Side-by-side
Incorporating Physical Constraints
Probabilistic Inverse Problem with Differentiable Simulations
Diffusion-based Time Prediction
Unconditional Stability
Graph-based Diffusion Models
Distributional Accuracy of Diffusion Graph Nets
Discussion of Probabilistic Learning

Reinforcement Learning

Introduction to Reinforcement Learning
Controlling Burgers’ Equation with Reinforcement Learning

Improved Gradients

Scale-Invariance and Inversion
Simple Example comparing Different Optimizers
Scale Invariant Physics Training
Learning to Invert Heat Conduction with Scale-invariant Updates
Half-Inverse Gradients
Coupled Oscillators with Half-Inverse Gradients
Discussion of Improved Gradients

Fast Forward Topics

Additional Topics
Model Reduction and Time Series
Unstructured Meshes and Meshless Methods
Generative Adversarial Networks

End Matter

Outlook
References
Notation and Abbreviations

Repository
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By N. Thuerey, B. Holzschuh, P. Holl, G. Kohl, M. Lino, Q. Liu, P. Schnell, F. Trost

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