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This research introduces CI-PINN, a novel architecture that embeds temporal causality directly into the neural network structure via Volterra-type integral terms to solve evolution equations. By addressing the inherent limitations of standard PINNs in handling time-dependent dynamics, the method achieves superior accuracy and robustness even with sparse data configurations. The work represents a significant advancement in machine learning for scientific computing, specifically enhancing the capability of AI models to respect physical laws over time.
CI-PINN: Causal Integral Physics-Informed Neural Network for Solving Evolution Equations
arXiv:2609.36615v1 Announce Type: cross 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.
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.
Numerical Analysis (math.NA); Machine Learning (cs.LG)
arXiv:2609.36615 [math.NA]
(or arXiv:2609.36615v1 [math.NA] for this version)
https://doi.org/10.48550/arXiv.2609.36615 Focus to learn more arXiv-issued DOI via DataCite (pending registration)
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