PENCO: A Physics-Energy-Numerics-Consistent Operator for 3D Phase Field Modeling

TL;DR

PENCO is a hybrid neural operator framework that integrates physical laws with data-driven methods to accurately and efficiently model 3D phase-field dynamics, outperforming existing neural operators in stability and data efficiency.

Contribution

The paper introduces PENCO, a novel physics-energy-numerics-consistent operator that combines physical constraints with neural operators for improved long-term accuracy in 3D phase-field modeling.

Findings

PENCO achieves higher accuracy than state-of-the-art neural operators.

PENCO demonstrates superior stability and data efficiency.

PENCO maintains physically consistent evolution in complex 3D benchmarks.

Abstract

Accurate and efficient solutions of spatiotemporal partial differential equations (PDEs), such as phase-field models, are fundamental for understanding interfacial dynamics and microstructural evolution in materials science and fluid mechanics. Neural operators (NOs) have recently emerged as powerful data-driven alternatives to traditional solvers; however, existing architectures often accumulate temporal errors, struggle to generalize over long temporal horizons, and require large training datasets. To overcome these limitations, we propose PENCO (Physics-Energy-Numerics-Consistent Operator), a hybrid operator-learning framework that integrates physical laws with data-driven neural operator methods, using either the Fourier Neural Operator (FNO-4D) or the Multi-Head Neural Operator (MHNO) architecture as the backbone. The formulation introduces an enhanced L^2 Gauss-Lobatto collocation residual around the temporal midpoint that robustly enforces the governing dynamics and significantly improves accuracy, a Fourier-space numerical consistency term that captures the balanced behavior of semi-implicit discretizations, and an energy-dissipation constraint that ensures thermodynamic consistency. Additional low-frequency spectral anchoring and teacher-consistency mechanisms further stabilize learning and suppress long-term error growth. This hybrid design enables PENCO to preserve governing physics while mitigating long-term error growth. Through extensive three-dimensional phase-field benchmarks on periodic cubic domains, covering phase ordering, crystallization, epitaxial growth, and complex pattern formation, PENCO demonstrates superior accuracy, stability, and data efficiency compared to state-of-the-art neural operators, including FNO-4D and MHNO, while maintaining physically consistent evolution. The associated dataset and implementation are available at github.com/MBamdad/PENCO.