[Paper Review] Physics-Guided Deep Learning for Dynamical Systems: A Survey

Rui Wang, Rose Yu|arXiv (Cornell University)|Jul 2, 2021

Model Reduction and Neural Networks44 citations

TL;DR

A comprehensive survey of physics-guided deep learning approaches for dynamical systems, detailing four problem types (solving differential equations, forecasting dynamics, learning residuals, and equation discovery) and four architectural/doctrine categories, with case studies like PINNs and TF-Net.

ABSTRACT

Modeling complex physical dynamics is a fundamental task in science and engineering. Traditional physics-based models are sample efficient, and interpretable but often rely on rigid assumptions. Furthermore, direct numerical approximation is usually computationally intensive, requiring significant computational resources and expertise, and many real-world systems do not have fully-known governing laws. While deep learning (DL) provides novel alternatives for efficiently recognizing complex patterns and emulating nonlinear dynamics, its predictions do not necessarily obey the governing laws of physical systems, nor do they generalize well across different systems. Thus, the study of physics-guided DL emerged and has gained great progress. Physics-guided DL aims to take the best from both physics-based modeling and state-of-the-art DL models to better solve scientific problems. In this paper, we provide a structured overview of existing methodologies of integrating prior physical knowledge or physics-based modeling into DL, with a special emphasis on learning dynamical systems. We also discuss the fundamental challenges and emerging opportunities in the area.

Motivation & Objective

• Motivate the integration of physical laws with deep learning to efficiently model complex dynamical systems.
• Classify and analyze methods that fuse physics with DL across loss-based, architectural, hybrid, and symmetry-aware approaches.
• Highlight challenges and opportunities in data efficiency, generalization, and equation discovery.
• Provide structured guidance through four core problem formulations relevant to dynamical systems.

Proposed method

• Formulates four main learning problems: solving differential equations, learning dynamics residuals, dynamics forecasting, and discovering governing equations.
• Reviews physics-guided loss functions (e.g., PINNs) that impose soft physical constraints during training.
• Discusses architecture design that enforces hard physical constraints via specialized neural modules.
• Covers hybrid physics-DL models that combine physics solvers with data-driven components.
• Explores invariant/equivariant DL models that respect physical symmetries to improve generalization.

Experimental results

Research questions

• RQ1 What are the main problem formulations in physics-guided deep learning for dynamical systems?
• RQ2 How do loss-based, architectural, hybrid, and symmetry-based methods leverage physical knowledge?
• RQ3 What are the benefits and limitations of physics-guided DL for stability, accuracy, and generalization?
• RQ4 How can governing equations be discovered from data with DL techniques?
• RQ5 What are emerging opportunities and challenges in this field?

Key findings

• Physics-guided DL can accelerate data simulation by substituting or augmenting numerical solvers with neural predictors.
• Incorporating physical constraints improves scientific validity and can enhance generalization across systems.
• Hard (architectural) and soft (loss-based) constraints offer complementary benefits and trade-offs in accuracy and robustness.
• Hamiltonian and Lagrangian neural nets enforce energy conservation properties in learned dynamics.
• Case studies show improved physical consistency, e.g., divergence-free velocity fields and energy spectrum alignment in turbulent flow prediction.