CI-PINN: Causal Integral Physics-Informed Neural Network for Solving 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.
Related
- Solving Fredholm Integral Equations with Physics-Informed Neural Networks: Single-and double-network approaches
- QGPINNs: A Physics-Informed Neural Network Framework for Nonlocal Differential Equations on Quantum Graphs
- AKAPINN: Adaptive Kolmogorov-Arnold Physics-Informed Neural Networks for approximating solutions to quasilinear partial differential equations
- Boosted Physics-Informed Neural Networks for Solving Homogeneous and Non-Homogeneous Lane-Emden Type Equations
- Comparison of a Parametric Physics-Informed Neural Network and a Tensorial Reduced-Order Model for the Shallow-Water Dam-Break Problem
BibTeX
@misc{feng2026cipinn,
title = {{CI-PINN: Causal Integral Physics-Informed Neural Network for Solving Evolution Equations}},
author = {Xiaodong Feng and Ziyu Sun and Tao Tang and Xiaoliang Wan and Tao Zhou},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.36615},
eprint = {2609.36615},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.36615},
}
RIS
TY - UNPB
TI - CI-PINN: Causal Integral Physics-Informed Neural Network for Solving Evolution Equations
AU - Xiaodong Feng
AU - Ziyu Sun
AU - Tao Tang
AU - Xiaoliang Wan
AU - Tao Zhou
PY - 2026
DA - 2026/09/29
PB - arXiv
DO - 10.48550/arxiv.2609.36615
UR - https://arxiv.org/abs/2609.36615
AB - 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.
ER -