Augmenting Physical Models with Deep Networks for Complex Dynamics Forecasting

Abstract: Forecasting complex dynamical phenomena in settings where only partial knowledge of their dynamics is available is a prevalent problem across various scientific fields. While purely data-driven approaches are arguably insufficient in this context, standard physical modeling based approaches tend to be over-simplistic, inducing non-negligible errors. In this work, we introduce the APHYNITY framework, a principled approach for augmenting incomplete physical dynamics described by differential equations with deep data-driven models. It consists in decomposing the dynamics into two components: a physical component accounting for the dynamics for which we have some prior knowledge, and a data-driven component accounting for errors of the physical model. The learning problem is carefully formulated such that the physical model explains as much of the data as possible, while the data-driven component only describes information that cannot be captured by the physical model, no more, no less. This not only provides the existence and uniqueness for this decomposition, but also ensures interpretability and benefits generalization. Experiments made on three important use cases, each representative of a different family of phenomena, i.e. reaction-diffusion equations, wave equations and the non-linear damped pendulum, show that APHYNITY can efficiently leverage approximate physical models to accurately forecast the evolution of the system and correctly identify relevant physical parameters. Code is available at https://github.com/yuan-yin/APHYNITY .

• The paper introduces the APHYNITY framework that fuses physical models with deep networks to enhance forecasting of complex dynamical systems.
• It decomposes system dynamics into a physical component and a data-driven residual, ensuring interpretability and minimal error.
• Experimental results show that APHYNITY achieves superior accuracy over standard models in predicting reaction-diffusion, wave phenomena, and non-linear systems.

Augmenting Physical Models with Deep Networks for Complex Dynamics Forecasting

The paper "Augmenting Physical Models with Deep Networks for Complex Dynamics Forecasting" presents the APHYNITY framework, an innovative approach for forecasting complex dynamical systems by combining incomplete physical models with deep learning techniques. The authors address the challenge of accurately predicting the evolution of dynamical systems where partial knowledge of their dynamics is available. This is a common problem in scientific fields such as environmental science, health sciences, and various industrial applications.

Overview and Methodology

The authors critique conventional model-based (MB) approaches, which rely heavily on partial or ordinary differential equations (PDEs/ODEs) based on a fundamental understanding of physical phenomena. Such models tend to oversimplify dynamics, resulting in substantial errors. Conversely, data-driven methods, predominantly powered by ML, often fail to extrapolate accurately due to their prior-agnostic nature. APHYNITY bridges this gap by decomposing the dynamics into two components: a physical part derived from MB models and a supplementary data-driven component to address errors and limitations inherent in the physical models.

The paper explores situations where physical models inadequately describe observed dynamics due to idealized assumptions, unknown external influences, or computational constraints. APHYNITY aims to enhance these models by leveraging the known dynamics and augmenting them with deep neural networks (DNNs) to model the residual dynamics. The framework is detailed, ensuring existence and uniqueness for the decomposition, which is vital for interpretability and generalization.

Theoretical Contributions

The authors provide robust theoretical support for the decomposition's uniqueness and existence under certain conditions. They emphasize that the decomposition into MB and ML components is not arbitrary; rather, it is carefully structured to ensure the minimal norm of the data-driven component while satisfying the fidelity to actual observed dynamics. The learning algorithm hinges on this decomposition scheme and employs an adaptive optimization strategy to ensure stable convergence, enabling precise forecasting and parameter identification.

Experimental Results

Significant experiments across three types of dynamical phenomena—reaction-diffusion equations, wave equations, and the non-linear damped pendulum—demonstrate the efficacy of APHYNITY. In each case, even with incomplete prior knowledge, the framework consistently delivered results closely aligning with those obtained from complete physical models. Notably, APHYNITY surpassed standard data-driven techniques by achieving higher forecasting accuracy and reducing errors in physical parameter identification.

Implications and Future Directions

The implications of this work are manifold. Practically, APHYNITY opens avenues for enhancing MB models in various fields where incomplete dynamics are already partially understood. Theoretically, it provides a foundation for further research into hybrid modeling approaches, potentially expanding into areas like partially observed settings and real-world applications such as climate modeling, robotics, and AI-driven reinforcement learning.

The findings suggest an ongoing trend toward integrating ML with MB paradigms, refining our ability to harness data-driven insights while maintaining grounding in established physical knowledge. Future research could explore the framework's extension to scenarios with dynamically changing parameters or further incorporation of domain-specific constraints and properties.

In conclusion, the APHYNITY framework exemplifies a sophisticated approach to complex dynamics forecasting, offering clear advantages in terms of adaptability, accuracy, and interpretability in hybrid modeling contexts. As AI progresses, such innovative methodologies are pivotal in addressing the nuanced challenges of dynamic system modeling.