Physics-Informed Neural Networks with Message-Passing Weights

Hyunwoo Cho, Hwijae Son, Hyung Ju Hwang

Keywords: Physics-informed Neural Networks, Adaptive Loss-Balancing Algorithms, causal PINN, Belief Propagation

Abstract:

Adaptive loss balancing algorithms play a crucial role in improving the performance of Physics-Informed Neural Networks (PINNs) by effectively managing the weights assigned to different loss components. Most notably, Wang et al. (2022) introduced Causal Physics-Informed Neural Networks (Causal PINNs), which achieve superior performance by simply reformulating the loss function based on the causal structure that emerges from time dependency. However, despite their empirical success, a solid theoretical analysis for the effectiveness of Causal PINNs has not received adequate attention. This paper addresses this gap by providing a theoretical rationale for Causal PINNs through the Belief Propagation (BP) algorithm, which is commonly used for causal inference. In addition, motivated by this analysis, we propose a Message Passing PINNs (MP-PINNs), a novel adaptive weighting algorithm. Through extensive numerical experiments, we demonstrate that the proposed MP-PINNs significantly outperform existing adaptive weighting methods, exhibiting superior performance in solving complex PDEs. Our findings highlight the potential of MP-PINNs as a powerful tool to enhance both the accuracy
and efficiency of PINNs.

Primary Area: neurosymbolic & hybrid AI systems (physics-informed, logic & formal reasoning, etc.)

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Submission Number: 14033