Causal integral networks improve solving evolution equations with physics
CI-PINN: Causal Integral Physics-Informed Neural Network for Solving Evolution Equations
Machine Learning
Summary
Solving equations that describe how things change over time can be tricky for computers. The usual methods don’t always remember how past events affect the future well enough. The authors created a special kind of neural network that better remembers past information in its design, helping it solve these time-dependent equations more accurately. Their method works especially well when only limited data points are available. It also doesn’t need much fine-tuning to perform well.
What this means in practice
- •For computational engineers: Simulate time-dependent physical systems more accurately with fewer data points by using causal integral physics-informed neural networks.
- •For software developers in scientific computing: Integrate causal integral neural network modules into existing PDE solvers to enhance temporal prediction without extra costly retraining.
Authors
Xiaodong Feng, Ziyu Sun, Tao Tang, Xiaoliang Wan, Tao Zhou
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.