# Physics-informed neural operator for parametric phase-field modelling of interfacial degradation and microstructural evolution

arXiv: https://arxiv.org/abs/2603.09693 · HTML: https://mlsanity.com/paper/2603.09693/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 26 of its 71 references that are in the map.
- First posted on arXiv 10 Mar 2026.

## Builds on (most cited first, 26 of 26)

- [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)
- [GradNorm: Gradient Normalization for Adaptive Loss Balancing in Deep Multitask Networks](https://mlsanity.com/paper/1711.02257/) (2017)
- [Scientific Machine Learning through Physics-Informed Neural Networks: Where we are and What's next](https://mlsanity.com/paper/2201.05624/) (2022)
- [Learning the solution operator of parametric partial differential equations with physics-informed DeepOnets](https://mlsanity.com/paper/2103.10974/) (2021)
- [Physics-Informed Neural Operator for Learning Partial Differential Equations](https://mlsanity.com/paper/2111.03794/) (2021)
- [hp-VPINNs: Variational Physics-Informed Neural Networks With Domain Decomposition](https://mlsanity.com/paper/2003.05385/) (2020)
- [Neural Operators for Accelerating Scientific Simulations and Design](https://mlsanity.com/paper/2309.15325/) (2023)
- [An Expert's Guide to Training Physics-informed Neural Networks](https://mlsanity.com/paper/2308.08468/) (2023)
- [Solving Allen-Cahn and Cahn-Hilliard Equations using the Adaptive Physics Informed Neural Networks](https://mlsanity.com/paper/2007.04542/) (2020)
- [Efficient training of physics-informed neural networks via importance sampling](https://mlsanity.com/paper/2104.12325/) (2021)
- [Learning two-phase microstructure evolution using neural operators and autoencoder architectures](https://mlsanity.com/paper/2204.07230/) (2022)
- [Towards Stability of Autoregressive Neural Operators](https://mlsanity.com/paper/2306.10619/) (2023)
- [Error analysis for physics informed neural networks (PINNs) approximating Kolmogorov PDEs](https://mlsanity.com/paper/2106.14473/) (2021)
- [Gradient Alignment in Physics-informed Neural Networks: A Second-Order Optimization Perspective](https://mlsanity.com/paper/2502.00604/) (2025)
- [Applications of physics informed neural operators](https://mlsanity.com/paper/2203.12634/) (2022)
- [Phase-Field DeepONet: Physics-informed deep operator neural network for fast simulations of pattern formation governed by gradient flows of free-energy functionals](https://mlsanity.com/paper/2302.13368/) (2023)
- [Variational Physics Informed Neural Networks: the role of quadratures and test functions](https://mlsanity.com/paper/2109.02035/) (2021)
- [Accelerating Phase Field Simulations Through a Hybrid Adaptive Fourier Neural Operator with U-Net Backbone](https://mlsanity.com/paper/2406.17119/) (2024)
- [Physics-consistent machine learning: output projection onto physical manifolds](https://mlsanity.com/paper/2502.15755/) (2025)
- [Sharp-PINNs: staggered hard-constrained physics-informed neural networks for phase field modelling of corrosion](https://mlsanity.com/paper/2502.11942/) (2025)
- [Equivariant U-Shaped Neural Operators for the Cahn-Hilliard Phase-Field Model](https://mlsanity.com/paper/2509.01293/) (2025)
- [A phase field formulation for dissolution-driven stress corrosion cracking](https://mlsanity.com/paper/2011.12068/) (2020)
- [Enforcing hidden physics in physics-informed neural networks](https://mlsanity.com/paper/2511.14348/) (2025)
- [A generalised, multi-phase-field theory for dissolution-driven stress corrosion cracking and hydrogen embrittlement](https://mlsanity.com/paper/2205.12096/) (2022)
- [Learning coupled Allen-Cahn and Cahn-Hilliard phase-field equations using Physics-informed neural operator(PINO)](https://mlsanity.com/paper/2507.18731/) (2025)

## Built on by (most cited first, 1 of 1)

- [On the training of physics-informed neural operators for solving parametric partial differential equations](https://mlsanity.com/paper/2606.06164/) (2026)

## Abstract (arXiv)

Predicting interfacial degradation and microstructural evolution through phase-field modelling is computationally intensive, particularly for parametric studies. While neural operators such as the Fourier neural operator (FNO) offer a promising route to acceleration, the absence of physical constraints compromises their generalisation and long-term stability. Here, we present PF-PINO, a physics-informed neural operator framework that embeds phase-field governing equation residuals directly into the training loss, together with gradient normalisation loss balancing and a staggered training scheme for coupled multi-field systems. PF-PINO is validated on four benchmarks including pencil-electrode corrosion, electro-polishing corrosion, dendritic solidification, and spinodal decomposition, collectively probing parametric generalisation across orders-of-magnitude kinetic regimes, non-periodic boundaries with stochastic morphologies, coupled multi-field dynamics with out-of-distribution extrapolation, and long-term autoregressive stability under complex pattern formation. Across all benchmarks, PF-PINO reduces relative L2 errors by up to an order of magnitude over FNO and achieves sub-mesh-size interface accuracy, establishing a data-efficient surrogate for high-throughput parametric phase-field assessment in computational materials science and engineering applications.