Embedding Physics-Informed Neural Networks into Numerical Schemes for Modeling Dynamical Systems

Abstract

Context and relevance. Physics-Informed Neural Networks (PINNs) are considered a promising tool for mathematical modeling of dynamical systems described by differential equations. However, classical PINN approaches require repeated computation of high-order derivatives, which leads to significant computational costs and limits their applicability in modeling tasks. Objective. To develop and experimentally validate an approach to mathematical modeling of dynamical systems based on embedding physics-informed neural networks into classical numerical schemes. Hypothesis. Embedding numerical schemes into the architecture of physics-informed neural networks can improve the computational efficiency of dynamical system modeling while maintaining accuracy comparable to classical PINN approaches. Methods and materials. A neural network architecture integrating a generalized θ-scheme (trapezoidal method) directly into the PINN architecture is proposed. A compact parametric network with 13 trainable parameters adaptively selects the balance between explicit and implicit numerical schemes at different points of the space–time domain. The experimental study was conducted on four types of differential equations characteristic of various dynamical processes: the heat equation, wave equation, reaction–diffusion equation, and Burgers equation. Results. In the experiments, under a fixed training time budget, the proposed 13-parameter model achieves comparable or higher accuracy than a 67-parameter classical PINN, while exhibiting significantly lower variance across runs. Conclusions. Integrating numerical schemes into the architecture of physics-informed neural networks improves the efficiency of constructing dynamical system models and provides interpretability of the architecture through the underlying numerical methods. The proposed approach can be considered a basis for building compact models in surrogate modeling tasks.