Multiphysics problems such as multicomponent diffusion, phase transformations in multiphase systems and alloy solidification involve numerical solution of a coupled system of nonlinear partial differential equations (PDEs). Numerical solutions of these PDEs using mesh-based methods require spatiotemporal discretization of these equations. Hence, the numerical solutions are often sensitive to discretization parameters and may have inaccuracies (resulting from grid-based approximations). Moreover, choice of finer mesh for higher accuracy make these methods computationally expensive. Neural network-based PDE solvers are emerging as robust alternatives to conventional numerical methods because these use machine learnable structures that are grid-independent, fast and accurate. However, neural network based solvers require large amount of training data, thus affecting their generalizabilty and scalability. These concerns become more acute for coupled systems of time-dependent PDEs. To address these issues, we develop a new neural network based framework that uses encoder-decoder based conditional Generative Adversarial Networks with ConvLSTM layers to solve a system of Cahn-Hilliard equations. These equations govern microstructural evolution of a ternary alloy undergoing spinodal decomposition when quenched inside a three-phase miscibility gap. We show that the trained models are mesh and scale-independent, thereby warranting application as effective neural operators.
翻译:多构件扩散、多相系统的阶段转换和合金固化等多物理问题,如多构件扩散、多相系统的阶段转换和合金固化,涉及非线性部分方程组合系统(PDEs)的数字解决方案。使用基于网格的方法,这些PDE的数值解决方案需要对这些方方程进行边际分解。因此,数字解决方案往往对离散参数十分敏感,并可能存在不准确性(基于网格的近似结果)。此外,选择更精细的网状网状网状网状网状网状使这些方法在计算上变得昂贵。基于神经网络的PDE解决方案正在成为常规数字方法的强有力替代品,因为这些系统使用不依赖网格、快速和准确的机器学习结构。然而,基于神经网络的解决方案需要大量的培训数据,从而影响其总体的兼容性和可缩放度。对于基于时间的PDEs(基于网格的近似近似)系统来说,这些关切变得更加尖锐。为了解决这些问题,我们开发了一个新的神经网络框架,它使用基于电解分解的自动的Aversal 网络,具有CLTM层次的系统,这些可学习的机器可替代结构结构结构,这些系统正在对正在演制成一个磁式的磁变式的系统进行。