Uncertainty-Aware Physics-Informed Neural Networks

• Novel UA-PINN framework integrating adaptive activation functions and uncertainty-aware self-training • Effective spectral bias mitigation via learnable activation functions capturing multi-scale features • Error reduction by 1-2 orders of magnitude on stiff PDE benchmarks (Burgers/Allen-Cahn/KdV) • Automated loss balancing through Monte Carlo Dropout-based pseudo-labeling Physics-Informed Neural Networks (PINNs) are a pothis studyrful paradigm for solving Partial Differential Equations (PDEs). Hothis studyver, they often struggle with multi-scale or stiff problems. These challenges arise from the "spectral bias" of neural networks and the training's sensitivity to loss term this studyights. This paper introduces the Uncertainty-Aware Physics-Informed Neural Network (UA-PINN) to address both challenges. the proposed framework integrates two key innovations: (1) an Adaptive Activation Function (AAF) with learnable parameters. It dynamically adjusts its shape to better represent high-frequency and multi-scale solution components, directly mitigating spectral bias. (2) An Uncertainty-aware Self-Training (ST) mechanism. It uses Monte Carlo Dropout to estimate model uncertainty and periodically identifies "high-confidence" points with low physical residual and low uncertainty. These points serve as pseudo-labels to regularize and stabilize the training, thus automating the complex task of loss this studyight balancing. this study evaluated UA-PINN on challenging benchmarks, including the Allen-Cahn equation, the shock-forming Burgers' equation, and the KdV equation with complex soliton interactions. Results show that the proposed method significantly outperforms standard PINNs in all test cases. Specifically, UA-PINN achieves relative L2 error reductions of 80.4% on the Allen-Cahn equation (from 9.19 × 10⁻³ to 1.80 × 10⁻³), 35.5% on the Burgers' equation (from 3.11 × 10⁻² to 2.00 × 10⁻²), and 4.6% on the KdV equation (from 4.03 × 10⁻³ to 3.85 × 10⁻³). Ablation studies confirm that both the AAF and self-training components contribute to these improvements, with their combination yielding synergistic benefits. It also more accurately captures high-gradient features like shocks and phase transitions. This work provides a more accurate and robust PDE solver. It also demonstrates the great potential of combining architectural innovations with smart training strategies to advance scientific machine learning.