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Solving partial differential equations (PDEs) with machine learning typically requires training a new neural network for every new equation. This optimization is slow. We introduce MetaColloc. It is an optimization-free and data-free…

In this work, we present a novel forward differential deep learning-based algorithm for solving high-dimensional nonlinear backward stochastic differential equations (BSDEs). Motivated by the fact that differential deep learning can…

A new deep-learning neural network architecture based on high-order weak approximation algorithms for stochastic differential equations (SDEs) is proposed. The architecture enables the efficient learning of martingales by deep learning…

In this paper, we present a deep learning-based numerical method for approximating high dimensional stochastic partial differential equations (SPDEs). At each time step, our method relies on a predictor-corrector procedure. More precisely,…

This work considers stochastic Galerkin approximations of linear elliptic partial differential equations (PDEs) with stochastic forcing terms and stochastic diffusion coefficients, that cannot be bounded uniformly away from zero and…

We present a semi-real-time algorithm for minimal-time optimal path planning based on optimal control theory, dynamic programming, and Hamilton-Jacobi (HJ) equations. Partial differential equation (PDE) based optimal path planning methods…

Using a combination of recurrent neural networks and signature methods from the rough paths theory we design efficient algorithms for solving parametric families of path dependent partial differential equations (PPDEs) that arise in pricing…

Optimal control problem is typically solved by first finding the value function through Hamilton-Jacobi equation (HJE) and then taking the minimizer of the Hamiltonian to obtain the control. In this work, instead of focusing on the value…

Hamilton-Jacobi (HJ) reachability analysis is a widely used method for ensuring the safety of robotic systems. Traditional approaches compute reachable sets by numerically solving an HJ Partial Differential Equation (PDE) over a grid, which…

We study the problem of optimal portfolio selection under stochastic volatility within a continuous time reinforcement learning framework with portfolio constraints. Exploration is modeled through entropy-regularized relaxed controls, where…

In this work we apply the Deep Galerkin Method (DGM) described in Sirignano and Spiliopoulos (2018) to solve a number of partial differential equations that arise in quantitative finance applications including option pricing, optimal…

We introduce a deep neural network based method for solving a class of elliptic partial differential equations. We approximate the solution of the PDE with a deep neural network which is trained under the guidance of a probabilistic…

Recent observations have been made that bridge splitting methods arising from optimization, to the Hopf and Lax formulas for Hamilton-Jacobi Equations with Hamiltonians $H(p)$. This has produced extremely fast algorithms in computing…

Applications in quantitative finance such as optimal trade execution, risk management of options, and optimal asset allocation involve the solution of high dimensional and nonlinear Partial Differential Equations (PDEs). The connection…

This paper presents a new methodology to craft navigation functions for nonlinear systems with stochastic uncertainty. The method relies on the transformation of the Hamilton-Jacobi-Bellman (HJB) equation into a linear partial differential…

Inverse problems are important mathematical problems that seek to recover model parameters from noisy data. Since inverse problems are often ill-posed, they require regularization or incorporation of prior information about the underlying…

We propose a partial differential-integral equation (PDE) framework for deep neural networks (DNNs) and their associated learning problem by taking the continuum limits of both network width and depth. The proposed model captures the…

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…

We propose novel connections between several neural network architectures and viscosity solutions of some Hamilton--Jacobi (HJ) partial differential equations (PDEs) whose Hamiltonian is convex and only depends on the spatial gradient of…

An adaptive direct collocation method is developed for solving optimal control problems constrained by parabolic partial differential equations. The partial differential equation is first reformulated in a variational setting, where the…