Related papers: Solving High-Dimensional PDEs with Latent Spectral…
Autoregressive next-step prediction models have become the de-facto standard for building data-driven neural solvers to forecast time-dependent partial differential equations (PDEs). Denoise training that is closely related to diffusion…
Accurately solving partial differential equations (PDEs) is essential across many scientific disciplines. However, high-fidelity solvers can be computationally prohibitive, motivating the development of reduced-order models (ROMs).…
This paper introduces a new approximation scheme for solving high-dimensional semilinear partial differential equations (PDEs) and backward stochastic differential equations (BSDEs). First, we decompose a target semilinear PDE (BSDE) into…
Partial Differential Equations (PDEs) are fundamental for modeling physical systems, yet solving them in a generic and efficient manner using machine learning-based approaches remains challenging due to limited multi-input and multi-scale…
Neural operators have achieved strong performance in learning solution operators of partial differential equations (PDEs), but their inherently continuous representations struggle to capture discontinuities and sharp transitions. Existing…
Linearized shallow neural networks that are constructed by fixing the hidden-layer parameters have recently shown strong performance in solving partial differential equations (PDEs). Such models, widely used in the random feature method…
In this article, we introduce and analyze a deep learning based approximation algorithm for SPDEs. Our approach employs neural networks to approximate the solutions of SPDEs along given realizations of the driving noise process. If applied…
Deep Learning Reduced Order Models (ROMs) are becoming increasingly popular as surrogate models for parametric partial differential equations (PDEs) due to their ability to handle high-dimensional data, approximate highly nonlinear…
It is a challenging topic in applied mathematics to solve high-dimensional nonlinear partial differential equations (PDEs). Standard approximation methods for nonlinear PDEs suffer under the curse of dimensionality (COD) in the sense that…
Solving partial differential equations (PDEs) using neural networks has become a central focus in scientific machine learning. Training neural networks for singularly perturbed problems is particularly challenging due to certain parameters…
The scientific computation methods development in conjunction with artificial intelligence technologies remains a hot research topic. Finding a balance between lightweight and accurate computations is a solid foundation for this direction.…
Large Language Models (LLMs) have achieved remarkable success across many applications, with Mixture of Experts (MoE) models demonstrating great potential. Compared to traditional dense models, MoEs achieve better performance with less…
In this work, we propose an Operator Learning (OpL) method for solving boundary value inverse problems in partial differential equations (PDEs), focusing on recovering diffusion coefficients from boundary data. Inspired by the classical…
High-dimensional partial differential equations (PDE) appear in a number of models from the financial industry, such as in derivative pricing models, credit valuation adjustment (CVA) models, or portfolio optimization models. The PDEs in…
We develop and evaluate a method for learning solution operators to nonlinear problems governed by partial differential equations (PDEs). The approach is based on a finite element discretization and aims at representing the solution…
We present a method for learning latent stochastic differential equations (SDEs) from high-dimensional time series data. Given a high-dimensional time series generated from a lower dimensional latent unknown It\^o process, the proposed…
Latent diffusion models (LDMs) power state-of-the-art high-resolution generative image models. LDMs learn the data distribution in the latent space of an autoencoder (AE) and produce images by mapping the generated latents into RGB image…
A state-of-the-art deep domain decomposition method (D3M) based on the variational principle is proposed for partial differential equations (PDEs). The solution of PDEs can be formulated as the solution of a constrained optimization…
In this paper, we evaluate the effectiveness of deep operator networks (DeepONets) in solving both forward and inverse problems of partial differential equations (PDEs) on unknown manifolds. By unknown manifolds, we identify the manifold by…
We propose a framework for training neural networks that are coupled with partial differential equations (PDEs) in a parallel computing environment. Unlike most distributed computing frameworks for deep neural networks, our focus is to…