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In this paper we introduce a multilevel Picard approximation algorithm for general semilinear parabolic PDEs with gradient-dependent nonlinearities whose coefficient functions do not need to be constant. We also provide a full convergence…
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…
Recently, it has been great interest in the development of methods for solving nonlinear differential equations directly. Here, it is shown an algorithm based on Pad\'e approximants for solving nonlinear partial differential equations…
We propose a Deep-Picard iteration framework for high-dimensional nonlinear space-time fractional diffusion equations.The method is based on a nonlinear fractional Feynman--Kac fixed-point formulation, which replaces direct discretization…
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…
This work develops a class of probabilistic algorithms for the numerical solution of nonlinear, time-dependent partial differential equations (PDEs). Current state-of-the-art PDE solvers treat the space- and time-dimensions separately,…
Reduced-order models have long been used to understand the behavior of nonlinear partial differential equations (PDEs). Naturally, reduced-order modeling techniques come at the price of computational accuracy for a decrease in computation…
Partial differential equations (PDEs) are among the most universal and parsimonious descriptions of natural physical laws, capturing a rich variety of phenomenology and multi-scale physics in a compact and symbolic representation. This…
We consider a system of semilinear partial differential equations (PDEs) with a nonlinearity depending on both the solution and its gradient. The Neumann boundary condition depends on the solution in a nonlinear manner. The uniform…
Equations governing physico-chemical processes are usually known at microscopic spatial scales, yet one suspects that there exist equations, e.g. in the form of Partial Differential Equations (PDEs), that can explain the system evolution at…
We present a lightweighted neural PDE representation to discover the hidden structure and predict the solution of different nonlinear PDEs. Our key idea is to leverage the prior of ``translational similarity'' of numerical PDE differential…
Many important problems in science and engineering require solving the so-called parametric partial differential equations (PDEs), i.e., PDEs with different physical parameters, boundary conditions, shapes of computation domains, etc.…
We propose a neural network-based meta-learning method to efficiently solve partial differential equation (PDE) problems. The proposed method is designed to meta-learn how to solve a wide variety of PDE problems, and uses the knowledge for…
We introduce and analyze a method of learning-informed parameter identification for partial differential equations (PDEs) in an all-at-once framework. The underlying PDE model is formulated in a rather general setting with three unknowns:…
Solving partial differential equations (PDEs) is the canonical approach for understanding the behavior of physical systems. However, large scale solutions of PDEs using state of the art discretization techniques remains an expensive…
One of the most challenging problems in applied mathematics is the approximate solution of nonlinear partial differential equations (PDEs) in high dimensions. Standard deterministic approximation methods like finite differences or finite…
Partial Differential Equations (PDEs) are central to modeling complex systems across physical, biological, and engineering domains, yet traditional numerical methods often struggle with high-dimensional or complex problems. Physics-Informed…
Physics-Informed Neural Networks (PINNs) are machine learning tools that approximate the solution of general partial differential equations (PDEs) by adding them in some form as terms of the loss/cost function of a Neural Network. Most…
Machine learning for scientific applications faces the challenge of limited data. We propose a framework that leverages a priori known physics to reduce overfitting when training on relatively small datasets. A deep neural network is…
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…