Physics-Informed Regularization for Domain-Agnostic Dynamical System Modeling (2410.06366v1)

Abstract: Learning complex physical dynamics purely from data is challenging due to the intrinsic properties of systems to be satisfied. Incorporating physics-informed priors, such as in Hamiltonian Neural Networks (HNNs), achieves high-precision modeling for energy-conservative systems. However, real-world systems often deviate from strict energy conservation and follow different physical priors. To address this, we present a framework that achieves high-precision modeling for a wide range of dynamical systems from the numerical aspect, by enforcing Time-Reversal Symmetry (TRS) via a novel regularization term. It helps preserve energies for conservative systems while serving as a strong inductive bias for non-conservative, reversible systems. While TRS is a domain-specific physical prior, we present the first theoretical proof that TRS loss can universally improve modeling accuracy by minimizing higher-order Taylor terms in ODE integration, which is numerically beneficial to various systems regardless of their properties, even for irreversible systems. By integrating the TRS loss within neural ordinary differential equation models, the proposed model TREAT demonstrates superior performance on diverse physical systems. It achieves a significant 11.5% MSE improvement in a challenging chaotic triple-pendulum scenario, underscoring TREAT's broad applicability and effectiveness.

• The paper introduces a novel physics-informed regularization based on Time-Reversal Symmetry (TRS) that aligns forward and reverse trajectories to improve numerical accuracy in dynamical system modeling.
• The paper demonstrates that the TRS framework achieves superior prediction performance across diverse datasets and both conservative and non-conservative dynamics, reducing mean squared error compared to state-of-the-art methods.
• The paper establishes that integrating TRS within a GraphODE framework enhances model flexibility and scalability, enabling reliable multi-agent system predictions in complex applications such as robotics and climate modeling.

Physics-Informed Regularization for Domain-Agnostic Dynamical System Modeling

The paper under review presents a framework for high-precision modeling of diverse dynamical systems, integrating a novel physics-informed regularization termed Time-Reversal Symmetry (TRS) into neural ordinary differential equation (ODE) models. This development addresses the challenge of learning complex physical dynamics purely from data without significant computational resource consumption.

Key Contributions

1. Enforcing Time-Reversal Symmetry (TRS): The framework introduces a self-supervised TRS regularization term that enhances the numerical accuracy of system modeling by aligning forward and reverse trajectories predicted by a neural network. This alignment theoretically minimizes higher-order Taylor expansion terms during ODE integration, which is beneficial across systems regardless of their physical properties.
2. Broad Applicability: Unlike methods limited to energy-conservative systems, this framework is designed to model both conservative and non-conservative dynamics without requiring prior knowledge of specific physical priors. The model, named TREAT (Time-Reversal Symmetry ODE), demonstrates superior applicability and effectiveness in various system scenarios.
3. Theoretical Proof of Numerical Benefit: The paper provides a theoretical foundation proving that TRS can universally improve modeling accuracy. This is achieved by minimizing higher-order terms, which implies better prediction of fine-grained dynamics and an enhanced ability to handle long-term predictions.
4. Empirical Validation: The authors validate their model across nine datasets, spanning both simulated and real-world systems, including chaotic systems. TREAT not only outpaces existing state-of-the-art approaches in terms of mean squared error (MSE) but also shows robustness against diverse physical scenarios.
5. Flexibility and Scalability: By integrating TRS into a GraphODE framework, the model not only accommodates single-agent systems but extends naturally to multi-agent settings where interactions among agents are complex and pivotal for accurate trajectory modeling.

Discussion and Implications

The incorporation of TRS as a regularization term is a strategic enhancement that bridges specific physical implications and general numerical benefits. Its ability to preserve energies for conservative systems while acting as an inductive bias for non-conservative, reversible systems makes it a versatile tool. Moreover, the theoretical insights contribute to understanding the underlying numerical advantages in dynamic system prediction tasks.

The framework's domain-agnostic nature enhances its appeal for scenarios where physical laws are either unknown or complex to apply directly. By minimizing higher-order Taylor terms, TREAT provides not only a numerically robust solution but also a flexible approach capable of adjusting to various system complexities.

Future Directions

The implications of this research are substantial, potentially influencing future developments in AI where accurate modeling of dynamical systems is crucial. This includes applications in robotics, physical simulations, and possibly extending to forecasting in domains like climate modeling or epidemiology. The alignment of numerical techniques with physical principles, as demonstrated by TRS, might inspire further integration of physics-informed machine learning models targeting broader and more complex system dynamics.

In conclusion, the paper presents a comprehensive and insightful exploration into the harmonization of neural networks with physics-informed constraints. While TRS itself is a broader physical prior, the methodology demonstrated here provides both a compelling case for its utility and a pathway for enhancing the accuracy and utility of dynamic system modeling in computational science.