cnn phase-field surrogate

Convolutional neural networks (CNNs) for surrogate modeling of high-dimensional partial differential equations (PDEs) suffer from prohibitive computational costs due to reliance on large-scale, high-fidelity numerical simulations. Method: We propose a cross-dimensional transfer learning framework featuring a novel hybrid-dimensional (d- and (d−1)-dimensional) joint training paradigm. It leverages approximate solutions of lower-dimensional PDEs to guide training of high-dimensional CNN surrogates, enabling knowledge transfer and error compensation. The architecture employs a fully convolutional encoder–decoder, multi-scale transfer mechanisms, and PDE-informed dimensionality-reduced data generation, augmented with uncertainty quantification. Contribution/Results: On multiphase flow benchmark problems, our method achieves higher accuracy than Monte Carlo methods using only a few times fewer simulation budgets. Forward inference is negligible in cost, dramatically improving the cost-effectiveness and practicality of PDE surrogates.

Phase-field simulations of liquid metal dealloying (LMD) incur prohibitive computational costs at large spatial and temporal scales. To address this, we propose a conditional-parameterized fully convolutional U-Net surrogate model that—uniquely—integrates convolutional self-attention with physics-informed padding, preserving translation invariance while enabling cross-system and cross-scale generalization and extrapolation. Trained exclusively on short-duration, small-domain simulations, the model accurately approximates complex microstructural evolution: achieving <5% relative error within the training domain and <10% error when extrapolating to larger domains and longer timescales. Computational speedup reaches 16,000×, reducing simulation time from weeks to seconds. The framework supports variable-step leapfrog prediction and transfer across multi-alloy systems. This work establishes a scalable, high-fidelity, data-driven paradigm for multiscale LMD modeling.

This work addresses the prohibitive computational cost of traditional phase-field simulations in large-scale liquid metal dealloying (LMD), which hinders efficient prediction of complex microstructure evolution. The authors propose the first fully convolutional, conditionally parameterized U-Net surrogate model, integrating physics-informed inpainting, convolutional self-attention mechanisms, and flood-fill correction. Coupled with a conditional diffusion model to generate physically consistent initial conditions, the framework enables large-scale spatiotemporal extrapolation of LMD phase-field dynamics without reliance on expensive numerical solvers for initialization. The method supports flexible time-step skipping and generalizes across multi-component alloys. Experimental results demonstrate relative errors below 5% for key physical quantities within the training domain and under 15% during long-term extrapolation, achieving up to a 36,000-fold speedup—reducing simulations that typically require weeks to mere seconds.

This study addresses the challenge of accurately capturing tail characteristics—corresponding to extreme samples—in solution field distributions when using neural network surrogates for uncertainty propagation. Taking the heat conduction equation as a benchmark, the work systematically evaluates the modeling capabilities of fully connected networks and DeepONets under both data-driven and physics-informed loss formulations, including weak-form residuals, with particular emphasis on tail prediction performance. The authors propose a method to identify extrapolative samples and find that fully connected networks trained with weak-form residual losses achieve superior accuracy under extreme inputs. Experimental results demonstrate that worst-case prediction errors for tail samples exceed those of the mean field by an order of magnitude, yet the proposed approach significantly enhances predictive accuracy for extreme scenarios on numerical datasets.

This work addresses the challenge of generalizing phase-field fracture modeling across arbitrary geometries throughout the entire process—from crack initiation to propagation—by proposing a hybrid integrated finite element–neural network (IFENN) solver. The method uniquely combines DeepOKAN and convolutional neural networks (CNNs) to separately model the initiation and propagation stages, while incorporating artificial boundary conditions to enhance far-field prediction accuracy. Requiring only a single physics-informed training on a reference geometry and leveraging a Gaussian-point sampling strategy, the approach substantially reduces offline computational costs. Numerical experiments demonstrate that the model achieves high accuracy and strong generalization capabilities on both seen and unseen geometries, significantly lowering computational resource demands.

This work addresses the high computational cost of traditional numerical methods for phase-field models, which arise from their multiscale and nonlinear nature requiring fine spatiotemporal discretization. The authors propose a novel approach that integrates convex–concave energy splitting with physics-informed learning, embedding energy dissipation constraints directly into neural operator training for the first time. They introduce a Reaction-Diffusion Neural Operator (RDNO) architecture tailored to reaction-diffusion equations and combine it with the Deep Ritz method to solve the associated variational problems. Demonstrated on both isotropic Allen–Cahn dynamics and anisotropic dendritic growth, the method exhibits superior generalization over purely data-driven models, achieves faster inference than conventional Fourier spectral methods, and rigorously preserves the energy dissipation property inherent to the physical system.

This work addresses the challenge of balancing accuracy and efficiency in forward prediction and parameter inversion for highly nonlinear, rapidly varying spatiotemporal dynamical systems—such as plasma turbulence—by proposing FI-Conv, a convolutional operator network based on the U-Net architecture. FI-Conv uniquely integrates ConvNeXt V2 blocks into the operator learning framework, substantially reducing computational complexity while preserving the ability to model high-frequency features. The method enables efficient autoregressive forward prediction and facilitates gradient-based parameter inversion without retraining. Evaluated on the Hasegawa–Wakatani turbulence model, FI-Conv achieves high-accuracy short-term state prediction (at t≈3), maintains statistical fidelity over long-term evolution (up to t≈100), and accurately recovers underlying PDE parameters from observational trajectory data.

In this work, a data-driven framework based on Phase-Field simulations data is proposed to highlight the capabilities of neural networks to ensure accurate low dimensionality reduction of simulated microstructural images and to provide time-series analysis. The dataset was indeed constructed from high-fidelity Phase-Field simulations. Analyses demonstrated that the association of auto-encoder neural networks and principal component analyses leads to ensure efficient and significant dimensionality reduction: 1/196 of reduction ratio with more than 80% of accuracy. These findings give insight to apply analyses on data from the latent dimension. Application of Long Short Term Memory (LSTM) neural networks showed the possibility of making next frame predictions; that makes possible the acceleration of Phase-Field simulation without the need of high computing resources. We discussed the application of such a framework on various areas of research. Different methods are proposed from the conducted analyses, in order to ensure dimensionality reduction, including auto-encoders, principal component analysis and Artificial Neural Networks, and time-series analysis, including LSTM and Gated Recurrent Unit (GRU).