# Morephy-Net: An Evolutionary Multi-objective Optimization for Replica-Exchange-based Physics-informed Neural Operator Learning Networks

arXiv: https://arxiv.org/abs/2509.00663 · HTML: https://mlsanity.com/paper/2509.00663/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 23 of its 76 references that are in the map.
- First posted on arXiv 31 Aug 2025.

## Builds on (most cited first, 23 of 23)

- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [Characterizing possible failure modes in physics-informed neural networks](https://mlsanity.com/paper/2109.01050/) (2021)
- [DeepXDE: A deep learning library for solving differential equations](https://mlsanity.com/paper/1907.04502/) (2019)
- [Learning the solution operator of parametric partial differential equations with physics-informed DeepOnets](https://mlsanity.com/paper/2103.10974/) (2021)
- [Physics-informed neural networks with hard constraints for inverse design](https://mlsanity.com/paper/2102.04626/) (2021)
- [Hidden Physics Models: Machine Learning of Nonlinear Partial Differential Equations](https://mlsanity.com/paper/1708.00588/) (2017)
- [Respecting causality is all you need for training physics-informed neural networks](https://mlsanity.com/paper/2203.07404/) (2022)
- [Non-convex Learning via Replica Exchange Stochastic Gradient MCMC](https://mlsanity.com/paper/2008.05367/) (2020)
- [A Contour Stochastic Gradient Langevin Dynamics Algorithm for Simulations of Multi-modal Distributions](https://mlsanity.com/paper/2010.09800/) (2020)
- [Data Complexity Estimates for Operator Learning](https://mlsanity.com/paper/2405.15992/) (2024)
- [NSGA-PINN: A Multi-Objective Optimization Method for Physics-Informed Neural Network Training](https://mlsanity.com/paper/2303.02219/) (2023)
- [Active operator learning with predictive uncertainty quantification for partial differential equations](https://mlsanity.com/paper/2503.03178/) (2025)
- [Stationary Density Estimation of Ito Diffusions Using Deep Learning](https://mlsanity.com/paper/2109.03992/) (2021)
- [Interacting Contour Stochastic Gradient Langevin Dynamics](https://mlsanity.com/paper/2202.09867/) (2022)
- [Learning Low-Dimensional Nonlinear Structures from High-Dimensional Noisy Data: An Integral Operator Approach](https://mlsanity.com/paper/2203.00126/) (2022)
- [Constrained Exploration via Reflected Replica Exchange Stochastic Gradient Langevin Dynamics](https://mlsanity.com/paper/2405.07839/) (2024)
- [Level set learning with pseudo-reversible neural networks for nonlinear dimension reduction in function approximation](https://mlsanity.com/paper/2112.01438/) (2021)
- [Bayesian Federated Learning with Hamiltonian Monte Carlo: Algorithm and Theory](https://mlsanity.com/paper/2407.06935/) (2024)
- [iPINNER: An Iterative Physics-Informed Neural Network with Ensemble Kalman Filter](https://mlsanity.com/paper/2506.00731/) (2025)
- [State-Space Kolmogorov Arnold Networks for Interpretable Nonlinear System Identification](https://mlsanity.com/paper/2506.16392/) (2025)
- [Symbolic Regression of Data-Driven Reduced Order Model Closures for Under-Resolved, Convection-Dominated Flows](https://mlsanity.com/paper/2502.04703/) (2025)
- [A Non-Dominated Sorting Evolutionary Algorithm Updating When Required](https://mlsanity.com/paper/2507.03864/) (2025)

## Built on by (most cited first, 3 of 3)

- [Muon with Spectral Guidance: Efficient Optimization for Scientific Machine Learning](https://mlsanity.com/paper/2602.16167/) (2026)
- [PIP2 Net: Physics-informed Partition Penalty Deep Operator Network](https://mlsanity.com/paper/2512.15086/) (2025)
- [fPINN-DeepONet: A Physics-Informed Operator Learning Framework for Multi-term Time-fractional Mixed Diffusion-wave Equations](https://mlsanity.com/paper/2605.16594/) (2026)

## Abstract (arXiv)

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.