CI-PINN: Causal Integral Physics-Informed Neural Network for Solving Evolution Equations

2026-09-28

AI Summary

This study addresses the accuracy limitations of conventional physics-informed neural networks (PINNs) in solving evolution equations due to the absence of explicit temporal dependency encoding. We propose the Causal Integral Physics-Informed Neural Network (CI-PINN), which, for the first time, introduces a Volterra-type causal integral module at the architectural level to aggregate historical features and encode temporal causality, thereby overcoming the constraints of relying solely on loss function optimization. By integrating causal integration with the PINN framework, the proposed method significantly outperforms multiple baselines in benchmark tests. Notably, CI-PINN demonstrates exceptional performance under sparse collocation point scenarios and exhibits robust insensitivity to hyperparameter variations. This work establishes an efficient new paradigm for the numerical simulation of evolution equations.

Abstract

Physics-informed neural networks (PINNs) solve partial differential equations (PDEs) by incorporating governing physical laws into the training loss. For evolution equations, however, their conventional pointwise space--time representation does not explicitly encode temporal dependence, which can hinder accurate prediction. To mitigate this limitation, this work proposes a novel neural architecture termed a causal integral neural network (CinNet). The core module of CinNet is a Volterra-type causal integral term, which aggregates historical features to encode temporal dependence, thereby incorporating temporal causality at the architectural level rather than through training-level modifications as in many existing methods. Building on CinNet, we further develop a causal integral physics-informed neural network (CI-PINN) for solving evolution equations. Extensive numerical experiments on benchmark evolution equations demonstrate that the presented method outperforms various baseline PINN variants in terms of solution accuracy, with pronounced superiority under sparse-collocation scenarios. Additional empirical analyses show that CI-PINN exhibits low sensitivity to hyperparameter choices, while ablation studies confirm the effectiveness of the proposed network components.

Problem

• Physics-Informed Neural Networks
• Evolution Equations
• Temporal Dependence
• Partial Differential Equations

Innovation

• Physics-Informed Neural Networks
• Causal Integral Network
• Volterra-type Integral
• Evolution Equations
• Temporal Causality