Causality-Aware Training of Physics-Informed Neural Networks for Solving Inverse Problems

Inverse Physics-Informed Neural Networks (inverse PINNs) offer a robust framework for solving inverse problems governed by partial differential equations (PDEs), particularly in scenarios with limited or noisy data. However, conventional inverse PINNs do not explicitly incorporate causality, which hinders their ability to capture the sequential dependencies inherent in physical systems. This study introduces Causal Inverse PINNs (CI-PINNs), a novel framework that integrates directional causality constraints across both temporal and spatial domains. Our approach leverages customized loss functions that adjust weights based on initial conditions, boundary conditions, and observed data, ensuring the model adheres to the system’s intrinsic causal structure. To evaluate CI-PINNs, we apply them to three representative inverse PDE problems, including an inverse problem involving the wave equation and inverse source problems for the parabolic and elliptic equations, each requiring distinct causal considerations. Experimental results demonstrate that CI-PINNs significantly improve accuracy and stability compared to conventional inverse PINNs by progressively enforcing causality-driven conditions and data consistency. This work underscores the potential of CI-PINNs to enhance robustness and reliability in solving complex inverse problems across diverse physical domains.