Morephy-Net: An Evolutionary Multi-objective Optimization for Replica-Exchange-based Physics-informed Neural Operator Learning Networks
Binghang Lu, Changhong Mou, Guang Lin
TL;DR
Morephy-Net tackles the challenge of learning parametric PDE solution operators in noisy regimes by unifying physics-informed operator learning with evolutionary multi-objective optimization and Bayesian posterior sampling. It replaces ad hoc loss weighting with a refined NSGA-III framework to separately optimize data/operator losses and physics residuals, and employs replica-exchange stochastic gradient Langevin dynamics to improve global exploration and enable principled uncertainty quantification. The approach is instantiated as a physics-informed DeepONet with a Fourier convolution layer, trained to yield diverse Pareto-optimal solutions and a calibrated predictive distribution from posterior samples. Empirical results on the Burgers equation and time-fractional diffusion-wave equations show that Morephy-Net consistently improves accuracy, robustness to noise, and uncertainty calibration over standard operator-learning baselines, with notable gains in forward and inverse problems. The framework offers practical impact for reliable scientific inference and data-assisted PDE modeling, and it opens avenues for data assimilation and turbulent-flow applications.
Abstract
We propose an evolutionary Multi-objective Optimization for Replica-Exchange-based Physics-informed operator-learning Networks (Morephy-Net) to solve parametric partial differential equations (PDEs) in noisy data regimes, for both forward prediction and inverse identification. Existing physics-informed neural networks and operator-learning models (e.g., DeepONets and Fourier neural operators) often face three coupled challenges: (i) balancing data/operator and physics residual losses, (ii) maintaining robustness under noisy or sparse observations, and (iii) providing reliable uncertainty quantification. Morephy-Net addresses these issues by integrating: (i) evolutionary multi-objective optimization that treats data/operator and physics residual terms as separate objectives and searches the Pareto front, thereby avoiding ad hoc loss weighting; (ii) replica-exchange stochastic gradient Langevin dynamics to enhance global exploration and stabilize training in non-convex landscapes; and (iii) Bayesian uncertainty quantification obtained from stochastic sampling. We validate Morephy-Net on representative forward and inverse problems, including the one-dimensional Burgers equation and the time-fractional mixed diffusion--wave equation. The results demonstrate consistent improvements in accuracy, noise robustness, and calibrated uncertainty estimates over standard operator-learning baselines.