An End-to-End Deep Learning Method for Solving Nonlocal Allen-Cahn and Cahn-Hilliard Phase-Field Models

• URL: http://arxiv.org/abs/2410.08914v1
• Date: Fri, 11 Oct 2024 15:41:47 GMT
• Title: An End-to-End Deep Learning Method for Solving Nonlocal Allen-Cahn and Cahn-Hilliard Phase-Field Models
• Authors: Yuwei Geng, Olena Burkovska, Lili Ju, Guannan Zhang, Max Gunzburger,
• Abstract summary: We propose an efficient end-to-end deep learning method for solving nonlocal Allen-Cahn (AC) and Cahn-Hilliard (CH) phase-field models. We tailor the architecture of the neural network by incorporating a nonlocal kernel as an input channel to the neural network model.
• Score: 11.92061406443778
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We propose an efficient end-to-end deep learning method for solving nonlocal Allen-Cahn (AC) and Cahn-Hilliard (CH) phase-field models. One motivation for this effort emanates from the fact that discretized partial differential equation-based AC or CH phase-field models result in diffuse interfaces between phases, with the only recourse for remediation is to severely refine the spatial grids in the vicinity of the true moving sharp interface whose width is determined by a grid-independent parameter that is substantially larger than the local grid size. In this work, we introduce non-mass conserving nonlocal AC or CH phase-field models with regular, logarithmic, or obstacle double-well potentials. Because of non-locality, some of these models feature totally sharp interfaces separating phases. The discretization of such models can lead to a transition between phases whose width is only a single grid cell wide. Another motivation is to use deep learning approaches to ameliorate the otherwise high cost of solving discretized nonlocal phase-field models. To this end, loss functions of the customized neural networks are defined using the residual of the fully discrete approximations of the AC or CH models, which results from applying a Fourier collocation method and a temporal semi-implicit approximation. To address the long-range interactions in the models, we tailor the architecture of the neural network by incorporating a nonlocal kernel as an input channel to the neural network model. We then provide the results of extensive computational experiments to illustrate the accuracy, structure-preserving properties, predictive capabilities, and cost reductions of the proposed method.

Related papers

• A domain decomposition-based autoregressive deep learning model for unsteady and nonlinear partial differential equations [2.7755345520127936]
  We propose a domain-decomposition-based deep learning (DL) framework, named CoMLSim, for accurately modeling unsteady and nonlinear partial differential equations (PDEs) The framework consists of two key components: (a) a convolutional neural network (CNN)-based autoencoder architecture and (b) an autoregressive model composed of fully connected layers.
  arXiv  Detail & Related papers  (2024-08-26T17:50:47Z)
• Gradient Flow Based Phase-Field Modeling Using Separable Neural Networks [1.2277343096128712]
  We propose a separable neural network-based approximation of the phase field in a minimizing movement scheme to solve a gradient flow problem. The proposed method outperforms the state-of-the-art machine learning methods for phase separation problems.
  arXiv  Detail & Related papers  (2024-05-09T21:53:27Z)
• Adaptive Multilevel Neural Networks for Parametric PDEs with Error Estimation [0.0]
  A neural network architecture is presented to solve high-dimensional parameter-dependent partial differential equations (pPDEs) It is constructed to map parameters of the model data to corresponding finite element solutions. It outputs a coarse grid solution and a series of corrections as produced in an adaptive finite element method (AFEM)
  arXiv  Detail & Related papers  (2024-03-19T11:34:40Z)
• Typical and atypical solutions in non-convex neural networks with discrete and continuous weights [2.7127628066830414]
  We study the binary and continuous negative-margin perceptrons as simple non-constrained network models learning random rules and associations. Both models exhibit subdominant minimizers which are extremely flat and wide. For both models, the generalization performance as a learning device is shown to be greatly improved by the existence of wide flat minimizers.
  arXiv  Detail & Related papers  (2023-04-26T23:34:40Z)
• D4FT: A Deep Learning Approach to Kohn-Sham Density Functional Theory [79.50644650795012]
  We propose a deep learning approach to solve Kohn-Sham Density Functional Theory (KS-DFT) We prove that such an approach has the same expressivity as the SCF method, yet reduces the computational complexity. In addition, we show that our approach enables us to explore more complex neural-based wave functions.
  arXiv  Detail & Related papers  (2023-03-01T10:38:10Z)
• Accelerated Solutions of Coupled Phase-Field Problems using Generative Adversarial Networks [0.0]
  We develop a new neural network based framework that uses encoder-decoder based conditional GeneLSTM layers to solve a system of Cahn-Hilliard microstructural equations. We show that the trained models are mesh and scale-independent, thereby warranting application as effective neural operators.
  arXiv  Detail & Related papers  (2022-11-22T08:32:22Z)
• Closed-form Continuous-Depth Models [99.40335716948101]
  Continuous-depth neural models rely on advanced numerical differential equation solvers. We present a new family of models, termed Closed-form Continuous-depth (CfC) networks, that are simple to describe and at least one order of magnitude faster.
  arXiv  Detail & Related papers  (2021-06-25T22:08:51Z)
• Identification of Probability weighted ARX models with arbitrary domains [75.91002178647165]
  PieceWise Affine models guarantees universal approximation, local linearity and equivalence to other classes of hybrid system. In this work, we focus on the identification of PieceWise Auto Regressive with eXogenous input models with arbitrary regions (NPWARX) The architecture is conceived following the Mixture of Expert concept, developed within the machine learning field.
  arXiv  Detail & Related papers  (2020-09-29T12:50:33Z)
• Multipole Graph Neural Operator for Parametric Partial Differential Equations [57.90284928158383]
  One of the main challenges in using deep learning-based methods for simulating physical systems is formulating physics-based data. We propose a novel multi-level graph neural network framework that captures interaction at all ranges with only linear complexity. Experiments confirm our multi-graph network learns discretization-invariant solution operators to PDEs and can be evaluated in linear time.
  arXiv  Detail & Related papers  (2020-06-16T21:56:22Z)
• Belief Propagation Reloaded: Learning BP-Layers for Labeling Problems [83.98774574197613]
  We take one of the simplest inference methods, a truncated max-product Belief propagation, and add what is necessary to make it a proper component of a deep learning model. This BP-Layer can be used as the final or an intermediate block in convolutional neural networks (CNNs) The model is applicable to a range of dense prediction problems, is well-trainable and provides parameter-efficient and robust solutions in stereo, optical flow and semantic segmentation.
  arXiv  Detail & Related papers  (2020-03-13T13:11:35Z)
• Kernel and Rich Regimes in Overparametrized Models [69.40899443842443]
  We show that gradient descent on overparametrized multilayer networks can induce rich implicit biases that are not RKHS norms. We also demonstrate this transition empirically for more complex matrix factorization models and multilayer non-linear networks.
  arXiv  Detail & Related papers  (2020-02-20T15:43:02Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.