Related papers: Finite Expression Method with TranNet-based Functi…
Designing efficient and accurate numerical solvers for high-dimensional partial differential equations (PDEs) remains a challenging and important topic in computational science and engineering, mainly due to the "curse of dimensionality" in…
Solving partial differential equations (PDEs) with highly oscillatory solutions on complex domains remains a challenging and important problem. High-frequency oscillations and intricate geometries often result in prohibitively expensive…
In this paper, we introduce a new finite expression method (FEX) to solve high-dimensional partial integro-differential equations (PIDEs). This approach builds upon the original FEX and its inherent advantages with new advances: 1) A novel…
Nonlinear dynamics is a pervasive phenomenon observed in scientific and engineering disciplines. However, the task of deriving analytical expressions to describe nonlinear dynamics from limited data remains challenging. In this paper, we…
Transfer learning for partial differential equations (PDEs) is to develop a pre-trained neural network that can be used to solve a wide class of PDEs. Existing transfer learning approaches require much information of the target PDEs such as…
We present two effective methods for solving high-dimensional partial differential equations (PDE) based on randomized neural networks. Motivated by the universal approximation property of this type of networks, both methods extend the…
Modeling stochastic differential equations (SDEs) is crucial for understanding complex dynamical systems in various scientific fields. Recent methods often employ neural network-based models, which typically represent SDEs through a…
Deep learning approaches for partial differential equations (PDEs) have received much attention in recent years due to their mesh-freeness and computational efficiency. However, most of the works so far have concentrated on time-dependent…
Partial differential equations (PDEs) underlie our understanding and prediction of natural phenomena across numerous fields, including physics, engineering, and finance. However, solving parametric PDEs is a complex task that necessitates…
There has been an arising trend of adopting deep learning methods to study partial differential equations (PDEs). In this paper, we introduce a deep recurrent framework for solving time-dependent PDEs without generating large scale data…
Neural operators aim to learn mappings between infinite-dimensional function spaces, but their performance often degrades on complex or irregular geometries due to the lack of geometry-aware representations. We propose the Finite Element…
We introduce a method that combines neural operators, physics-informed machine learning, and standard numerical methods for solving PDEs. The proposed approach extends each of the aforementioned methods and unifies them within a single…
Deep learning has emerged as a compelling framework for scientific and engineering computing, motivating growing interest in neural network-based solvers for partial differential equations (PDEs). Within this landscape, network…
Neural network based methods have emerged as a promising paradigm for scientific computing, yet they face critical bottlenecks in high frequency function approximation and partial differential equation (PDE) solving.
High-dimensional partial differential equations (PDEs) pose significant challenges for numerical computation due to the curse of dimensionality, which limits the applicability of traditional mesh-based methods. Since 2017, the Deep BSDE…
We propose a method combining boundary integral equations and neural networks (BINet) to solve partial differential equations (PDEs) in both bounded and unbounded domains. Unlike existing solutions that directly operate over original PDEs,…
In this article I present a fast and direct method for solving several types of linear finite difference equations (FDE) with constant coefficients. The method is based on a polynomial form of the translation operator and its inverse, and…
We propose a finite-dimensional control-based method to approximate solution operators for evolutional partial differential equations (PDEs), particularly in high-dimensions. By employing a general reduced-order model, such as a deep neural…
Recent works have shown that deep neural networks can be employed to solve partial differential equations, giving rise to the framework of physics informed neural networks. We introduce a generalization for these methods that manifests as a…
The least squares method with deep neural networks as function parametrization has been applied to solve certain high-dimensional partial differential equations (PDEs) successfully; however, its convergence is slow and might not be…