Physics-based Deep Learning (Free PDF)

 

Physics-based Deep Learning: Combining Deep Learning with Physical Simulations

Physics-based Deep Learning (PBDL) is a practical learning resource that explores how deep learning can be combined with physical models and numerical simulations. The work is authored by Nils Thuerey, Benjamin Holzschuh, Philipp Holl, Georg Kohl, Mario Lino, Qiang Liu, Patrick Schnell, and Felix Trost. The current arXiv version is v4, revised in March 2025.

Unlike a traditional deep learning resource that focuses primarily on image, text, or tabular datasets, this book focuses on physical simulations and scientific computing. It explores how neural networks can work together with existing knowledge about physical systems instead of treating the problem as purely data-driven.

Download the PDF for free: https://arxiv.org/abs/2109.05237

What Is Physics-based Deep Learning?

Physics-based Deep Learning brings together two major areas:

• Deep Learning — neural networks learn patterns from data.

• Physics-based Modeling — physical laws and numerical simulations describe how real systems behave.

The central idea is not simply to replace traditional simulation methods with neural networks. Instead, the goal is to find useful ways to combine learned models with physical knowledge and numerical techniques.

This creates a hybrid approach where machine learning can help accelerate simulations, estimate physical states, solve difficult inverse problems, or incorporate physical constraints into the learning process.

Why Combine Physics and Deep Learning?

Traditional physical simulations can be computationally expensive, particularly when a problem must be solved repeatedly.

For example, simulations involving:

• Fluid flows

• Heat transfer

• Physical dynamics

• Engineering systems

• Environmental processes

• Complex scientific phenomena

may require substantial computational resources.

A neural network can potentially learn a specialized approximation for a particular problem domain. Once trained, that learned model can support repeated simulations much more efficiently in suitable scenarios. The PBDL authors describe this as an opportunity to combine specialized neural networks with established numerical solvers rather than discarding traditional simulation methods.

A Hands-on Learning Approach

One of the most interesting aspects of the resource is its practical focus.

The authors describe it as a hands-on and comprehensive guide, with concepts accompanied by interactive Jupyter notebooks. This makes the material particularly useful for learners who want to experiment with the methods rather than only read about them.

The notebooks allow readers to explore concepts computationally and understand how deep learning interacts with physical simulations.

Data-Driven Physics

One approach is to use data generated by real or simulated physical systems.

In this setting, the physical simulator can produce training data, while the neural network learns relationships from that data.

The important distinction is that the physical model provides the source of information, but the learning process itself may not directly enforce the physical rules.

This approach can be useful when large amounts of simulation data are available and a learned approximation is valuable.

Physical Loss Constraints

A more tightly connected approach incorporates physical knowledge into the learning objective.

Instead of asking a neural network to simply reproduce training examples, the learning process can also account for whether its predictions are consistent with relevant physical behavior.

This idea is closely related to physics-informed learning, where physical constraints help guide the training process.

The benefit is that the model does not have to rely entirely on patterns found in data. Physical knowledge can become part of the learning process.

Differentiable Physics

Another major topic is differentiable simulation.

A differentiable simulator allows information about how a simulation changes to flow through the learning process. This creates a much closer interaction between neural networks and numerical simulation.

Instead of treating the simulator as a completely separate black box, the learning algorithm can interact with it during optimization.

The PBDL material describes these approaches as an especially tight integration between deep learning and physical simulations.

Forward and Inverse Problems

Physics-based Deep Learning also considers two important types of scientific problems.

Forward Problems

A forward problem starts with known physical parameters or conditions and attempts to predict what happens next.

For example, a model might predict how a physical system evolves over time.

Inverse Problems

An inverse problem works in the opposite direction.

Instead of starting with known parameters and predicting observations, the goal is to use observations to determine unknown properties of the physical system.

Deep learning can be particularly useful for these problems because neural networks can learn complex relationships between observations and underlying physical parameters.

Reinforcement Learning for Physical Systems

The resource also explores reinforcement learning in the context of physical simulations.

In reinforcement learning, an agent learns by interacting with an environment and receiving feedback.

When combined with physics-based environments, this can be used to investigate problems involving:

• Control

• Optimization

• Physical decision-making

• Dynamic systems

• Simulation-based learning

This creates a connection between scientific simulation and intelligent control systems.

Generative AI and Physical Simulation

The newer PBDL v0.3 release adds a major section on generative AI, including diffusion-based approaches and physics-based constraints. The authors describe this as a substantial new chapter in the updated version.

This is particularly interesting because generative models can produce possible physical states or system behaviors, while physics-based constraints can help guide those predictions toward physically meaningful results.

It represents an emerging direction where generative AI is not used only for images or text but also for scientific and physical modeling.

Scientific Foundation Models

The authors connect these developments to the broader idea of scientific foundation models.

Traditional foundation models have largely focused on domains such as language, vision, and multimodal data. Physics-based learning introduces the possibility of models that can work with scientific systems while incorporating physical knowledge.

This could become important for areas where predictions need to respect the behavior of the underlying physical system.

Practical Applications

Physics-based Deep Learning can be relevant to a wide range of scientific and engineering problems.

Potential applications include:

• Fluid simulation

• Weather and environmental modeling

• Engineering design

• Computational physics

• Physical system control

• Surrogate modeling

• Inverse problems

• Scientific machine learning

• Simulation acceleration

The official project materials specifically position PBDL around combinations of physical modeling, numerical simulation, and neural-network-based learning.

What Makes This Resource Different?

The biggest difference is its focus on combining, rather than replacing.

The goal is not simply:

Physics → replaced by AI

Instead, the broader philosophy is:

Physics + Numerical Methods + Deep Learning

This is important because established numerical methods already contain decades of knowledge about physical systems. Combining them with machine learning can create specialized computational approaches while preserving valuable physical information.

Who Should Read It?

This resource is particularly suitable for:

• Deep learning students

• Physics students

• Computational scientists

• Scientific machine learning researchers

• Engineers

• Researchers working with simulations

• ML practitioners interested in scientific applications

• Students interested in differentiable physics

It is more specialized than a general deep learning book, so readers will benefit from having some background in machine learning and numerical or physical simulation concepts.

Strengths

1. Strong Practical Focus

The resource provides interactive Jupyter notebooks alongside many concepts.

2. Combines Multiple Fields

It connects deep learning, numerical simulation, physics, optimization, and scientific computing.

3. Covers Modern Topics

The current version includes topics such as differentiable physics, reinforcement learning, uncertainty modeling, and generative AI.

4. Useful for Scientific ML

It provides a strong conceptual foundation for understanding how machine learning can be applied to physical simulation problems.

Limitations

This is not a beginner-level introduction to deep learning.

The authors explicitly position the resource as a guide to deep learning in the context of physical simulations rather than an in-depth introduction to basic deep learning or numerical simulation.

Therefore, beginners may need additional resources to learn fundamental neural networks, optimization, and simulation concepts before working through the more advanced material.

Download the PDF for free: https://arxiv.org/abs/2109.05237

Final Verdict

Physics-based Deep Learning is an excellent resource for understanding the intersection of deep learning, physics, and computational simulation.

Its most valuable idea is that AI does not necessarily have to operate independently from scientific knowledge. Neural networks can work alongside numerical methods, physical constraints, and differentiable simulations to create new approaches to scientific computing.

The addition of generative AI in the latest version also makes the resource especially relevant to the rapidly developing field of scientific AI.