Lightweight Explainable Physics-Informed Neural Networks by Learnable Activation Function

Physics-informed neural networks (PINNs) provide a data-efficient framework for solving partial differential equations, but improving their accuracy often requires larger multilayer perceptron (MLP) backbones, increasing parameter count and computational cost. We investigate whether learnable activation functions can improve PINN performance while preserving lightweight backbones and explainable nonlinearities. We adapt the Cannistraci-Muscoloni-Gu generalized logistic-logit function (CMG) to PINNs and introduce CMG-GELU, which replaces the tanh term in the GELU approximation with learnable CMG. To support stable PINN training, we reformulate CMG's implicit logit phase into an explicit differentiable approximation and adopt a layer-wise parameterization that adds only two trainable activation parameters per hidden layer. We compare CMG and CMG-GELU with nine representative activation and architecture baselines across 25 forward and inverse PINN tasks using five random seeds. Compared with representative PINN method studies, our 25-task suite covers substantially more tasks, enabling a broader cross-task evaluation. CMG-GELU delivers the strongest aggregate performance and the highest cross-task consistency among the 11 methods, achieving the best mean rank, rank standard deviation, and Robustness Performance Ranking Index (RPRI), while CMG ranks second by RPRI. Learned activation parameters reveal task- and layer-dependent activation modulation patterns, providing parameter-level explainability, while low-dimensional parameter-profile embeddings provide exploratory task-level evidence of local organization among tasks with related physical regimes. These results support CMG-based learnable activations as a lightweight and explainable approach to improving PINNs across heterogeneous problems.