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

arXiv: https://arxiv.org/abs/2609.36615 · HTML: https://mlsanity.com/paper/2609.36615/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 16 of its 30 references that are in the map.
- First posted on arXiv 29 Sep 2026.

## Builds on (most cited first, 16 of 16)

- [Characterizing possible failure modes in physics-informed neural networks](https://mlsanity.com/paper/2109.01050/) (2021)
- [Solving Allen-Cahn and Cahn-Hilliard Equations using the Adaptive Physics Informed Neural Networks](https://mlsanity.com/paper/2007.04542/) (2020)
- [PPINN: Parareal Physics-Informed Neural Network for time-dependent PDEs](https://mlsanity.com/paper/1909.10145/) (2019)
- [PhyCRNet: Physics-informed Convolutional-Recurrent Network for Solving Spatiotemporal PDEs](https://mlsanity.com/paper/2106.14103/) (2021)
- [A Method for Representing Periodic Functions and Enforcing Exactly Periodic Boundary Conditions with Deep Neural Networks](https://mlsanity.com/paper/2007.07442/) (2020)
- [Mitigating Propagation Failures in Physics-informed Neural Networks using Retain-Resample-Release (R3) Sampling](https://mlsanity.com/paper/2207.02338/) (2022)
- [PINNsFormer: A Transformer-Based Framework For Physics-Informed Neural Networks](https://mlsanity.com/paper/2307.11833/) (2023)
- [Neural Galerkin Schemes with Active Learning for High-Dimensional Evolution Equations](https://mlsanity.com/paper/2203.01360/) (2022)
- [Evolutional Deep Neural Network](https://mlsanity.com/paper/2103.09959/) (2021)
- [A unified scalable framework for causal sweeping strategies for Physics-Informed Neural Networks (PINNs) and their temporal decompositions](https://mlsanity.com/paper/2302.14227/) (2023)
- [Sub-Sequential Physics-Informed Learning with State Space Model](https://mlsanity.com/paper/2502.00318/) (2025)
- [Exact Enforcement of Temporal Continuity in Sequential Physics-Informed Neural Networks](https://mlsanity.com/paper/2403.03223/) (2024)
- [Enforcing hidden physics in physics-informed neural networks](https://mlsanity.com/paper/2511.14348/) (2025)
- [A hybrid FEM-PINN method for time-dependent partial differential equations](https://mlsanity.com/paper/2409.02810/) (2024)
- [Integral regularization PINNs for evolution equations](https://mlsanity.com/paper/2503.23729/) (2025)
- [PIANO: Physics Informed Autoregressive Network](https://mlsanity.com/paper/2508.16235/) (2025)

## Abstract (arXiv)

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.