A Deep Neural Network Surrogate for High-Dimensional Random Partial Differential Equations
Developing efficient numerical algorithms for the solution of high dimensional random Partial Differential Equations (PDEs) has been a challenging task due to the well-known curse of dimensionality. We present a new solution framework for these problems based on a deep learning approach. Specifically, the random PDE is approximated by a feed-forward fully-connected deep residual network, with either strong or weak enforcement of initial and boundary constraints. The framework is mesh-free, and can handle irregular computational domains. Parameters of the approximating deep neural network are determined iteratively using variants of the Stochastic Gradient Descent (SGD) algorithm. The satisfactory accuracy of the proposed frameworks is numerically demonstrated on diffusion and heat conduction problems, in comparison with the converged Monte Carlo-based finite element results.
Code (0)
등록된 구현이 없습니다.
Similar Papers 제목 키워드 기반
Monte Carlo PINNs: deep learning approach for forward and inverse problems involving high dimensional fractional partial differential equations
We introduce a sampling based machine learning approach, Monte Carlo physics informed neural networks (MC-PINNs), for solving forward and inverse fractional partial differential equations (FPDEs). As a generalization of …
Neuro-symbolic partial differential equation solver
We present a highly scalable strategy for developing mesh-free neuro-symbolic partial differential equation solvers from existing numerical discretizations found in scientific computing. This strategy is unique in that i…
Deep UQ: Learning deep neural network surrogate models for high dimensional uncertainty quantification
State-of-the-art computer codes for simulating real physical systems are often characterized by a vast number of input parameters. Performing uncertainty quantification (UQ) tasks with Monte Carlo (MC) methods is almost …
Uncertainty QuantificationAnant-Net: Breaking the Curse of Dimensionality with Scalable and Interpretable Neural Surrogate for High-Dimensional PDEs
High-dimensional partial differential equations (PDEs) arise in diverse scientific and engineering applications but remain computationally intractable due to the curse of dimensionality. Traditional numerical methods str…
GPUKolmogorov-Arnold NetworksNeural network surrogates with uncertainty quantification for inverse problems in partial differential equations
Inverse problems for differential equations arise throughout science and engineering, where one seeks to infer unknown model parameters from noisy or incomplete observations. Traditional numerical methods for these probl…
Bayesian Inference