📄 Executive Summary
This textbook provides an applied guide to physics-based deep learning, also known as scientific machine learning, which blends numerical physical simulations with modern artificial intelligence architectures. Designed for researchers, engineers, and students with a working knowledge of deep learning fundamentals and basic partial differential equations, the book focuses on practical implementations using frameworks such as PyTorch, JAX, and the differentiable simulation library ΦFlow. The core scope centers on integrating domain knowledge from governing differential equations into neural network training pipelines to solve both forward simulation and inverse parameter estimation problems.
The text begins by establishing foundational numerical methods and optimization theory, demonstrating classical forward solvers for Burgers and Navier-Stokes equations alongside standard optimization techniques. It then introduces supervised surrogate modeling and neural operators, analyzing how network architectures like convolutional networks, Fourier neural operators, and transformers address local versus global receptive field dependencies. From there, the material progresses to soft physical constraints via physics-informed neural networks and residual formulations, highlighting their strengths in inverse problems and their limitations regarding convergence. The book then presents differentiable physics, demonstrating how embedding differentiable numerical solvers directly into training loops mitigates distribution shift and produces hybrid solver-in-the-loop systems.
Expanding beyond deterministic predictions, the textbook examines probabilistic modeling and simulation-based inference to capture physical uncertainties and multimodal behaviors. Readers learn to construct normalizing flows, continuous neural differential equations, score-based models, denoising diffusion probabilistic models, and flow matching frameworks guided by physical priors and graph representations. The text also contrasts differentiable physics with model-free reinforcement learning algorithms such as proximal policy optimization, and details advanced gradient inversion techniques—including scale-invariant physics and half-inverse gradients—to correct optimization scaling issues. While broad topics like reduced-order modeling, generative adversarial networks, and Lagrangian particle dynamics are treated as higher-level extensions pointing to external codebases, readers complete the book equipped to design, implement, and stabilize hybrid machine learning systems that interact directly with physical simulators.
