# Phase-Field DeepONet: Physics-informed deep operator neural network for fast simulations of pattern formation governed by gradient flows of free-energy functionals

arXiv: https://arxiv.org/abs/2302.13368 · HTML: https://mlsanity.com/paper/2302.13368/

## Key facts

- Cited by 26 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 26 Feb 2023.

## Built on by (most cited first, 26 of 26)

- [A Framework Based on Symbolic Regression Coupled with eXtended Physics-Informed Neural Networks for Gray-Box Learning of Equations of Motion from Data](https://mlsanity.com/paper/2305.10706/) (2023)
- [Basis-to-Basis Operator Learning Using Function Encoders](https://mlsanity.com/paper/2410.00171/) (2024)
- [Diffeomorphism Neural Operator for various domains and parameters of partial differential equations](https://mlsanity.com/paper/2402.12475/) (2024)
- [A finite element-based physics-informed operator learning framework for spatiotemporal partial differential equations on arbitrary domains](https://mlsanity.com/paper/2405.12465/) (2024)
- [Predicting Crack Nucleation and Propagation in Brittle Materials Using Deep Operator Networks with Diverse Trunk Architectures](https://mlsanity.com/paper/2501.00016/) (2024)
- [Fast meta-solvers for 3D complex-shape scatterers using neural operators trained on a non-scattering problem](https://mlsanity.com/paper/2405.12380/) (2024)
- [Geometry-Informed Neural Operator Transformer](https://mlsanity.com/paper/2504.19452/) (2025)
- [A physics-informed transformer neural operator for learning generalized solutions of initial boundary value problems](https://mlsanity.com/paper/2412.09009/) (2024)
- [A Physics-Informed Meta-Learning Framework for the Continuous Solution of Parametric PDEs on Arbitrary Geometries](https://mlsanity.com/paper/2504.02459/) (2025)
- [Integration of physics-informed operator learning and finite element method for parametric learning of partial differential equations](https://mlsanity.com/paper/2401.02363/) (2024)
- [Fluids You Can Trust: Property-Preserving Operator Learning for Incompressible Flows](https://mlsanity.com/paper/2602.15472/) (2026)
- [An End-to-End Deep Learning Method for Solving Nonlocal Allen-Cahn and Cahn-Hilliard Phase-Field Models](https://mlsanity.com/paper/2410.08914/) (2024)
- [Learn Singularly Perturbed Solutions via Homotopy Dynamics](https://mlsanity.com/paper/2502.00488/) (2025)
- [GALDS: A Graph-Autoencoder-based Latent Dynamics Surrogate model to predict neurite material transport](https://mlsanity.com/paper/2507.10871/) (2025)
- [Gradient Flow Based Phase-Field Modeling Using Separable Neural Networks](https://mlsanity.com/paper/2405.06119/) (2024)
- [DeepONet for Solving Nonlinear Partial Differential Equations with Physics-Informed Training](https://mlsanity.com/paper/2410.04344/) (2024)
- [Physics-Informed Deep B-Spline Networks](https://mlsanity.com/paper/2503.16777/) (2025)
- [Physics-informed neural operator for parametric phase-field modelling of interfacial degradation and microstructural evolution](https://mlsanity.com/paper/2603.09693/) (2026)
- [Structure-Aware Variational Learning of a Class of Generalized Diffusions](https://mlsanity.com/paper/2604.20188/) (2026)
- [Predicting Time-Dependent Flow Over Complex Geometries Using Operator Networks](https://mlsanity.com/paper/2512.04434/) (2025)
- [Intrinsic-Metric Physics-Informed Neural Networks (IM-PINN) for Reaction-Diffusion Dynamics on Complex Riemannian Manifolds](https://mlsanity.com/paper/2601.00834/) (2025)
- [Tackling multiphysics problems via finite element-guided physics-informed operator learning](https://mlsanity.com/paper/2603.01420/) (2026)
- [General Explicit Network (GEN): A novel deep learning architecture for solving partial differential equations](https://mlsanity.com/paper/2604.03321/) (2026)
- [DeepRitzSplit Neural Operator for Phase-Field Models via Energy Splitting](https://mlsanity.com/paper/2604.18261/) (2026)
- [SpectONet: A Physics-Guided Spectral Deep Operator Network for Euler-Bernoulli Beam Dynamics](https://mlsanity.com/paper/2607.25790/) (2026)
- [Theory Guided and Interpretable Neural Operator Design for Partial Differential Equation Learning](https://mlsanity.com/paper/2609.33715/) (2026)

## Abstract (arXiv)

Recent advances in scientific machine learning have shed light on the modeling of pattern-forming systems. However, simulations of real patterns still incur significant computational costs, which could be alleviated by leveraging large image datasets. Physics-informed machine learning and operator learning are two new emerging and promising concepts for this application. Here, we propose "Phase-Field DeepONet", a physics-informed operator neural network framework that predicts the dynamic responses of systems governed by gradient flows of free-energy functionals. Examples used to validate the feasibility and accuracy of the method include the Allen-Cahn and Cahn-Hilliard equations, as special cases of reactive phase-field models for nonequilibrium thermodynamics of chemical mixtures. This is achieved by incorporating the minimizing movement scheme into the framework, which optimizes and controls how the total free energy of a system evolves, instead of solving the governing equations directly. The trained operator neural networks can work as explicit time-steppers that take the current state as the input and output the next state. This could potentially facilitate fast real-time predictions of pattern-forming dynamical systems, such as phase-separating Li-ion batteries, emulsions, colloidal displays, or biological patterns.