Related papers: Error bounds for PDE-regularized learning

Verification of Neural Networks (NNs) that approximate the solution of Partial Differential Equations (PDEs) is a major milestone towards enhancing their trustworthiness and accelerating their deployment, especially for safety-critical…

Neural Controlled Differential Equations (NCDEs) are a state-of-the-art tool for supervised learning with irregularly sampled time series (Kidger, 2020). However, no theoretical analysis of their performance has been provided yet, and it…

In this paper, we study the dynamics of temporal difference learning with neural network-based value function approximation over a general state space, namely, \emph{Neural TD learning}. We consider two practically used algorithms,…

Physics informed neural networks approximate solutions of PDEs by minimizing pointwise residuals. We derive rigorous bounds on the error, incurred by PINNs in approximating the solutions of a large class of linear parabolic PDEs, namely…

The phenomenon of data distribution evolving over time has been observed in a range of applications, calling the needs of adaptive learning algorithms. We thus study the problem of supervised gradual domain adaptation, where labeled data…

We consider a linear-quadratic pde constrained optimal control problem on an evolving surface with pointwise state constraints. We reformulate the optimization problem on a fixed surface and approximate the reformulated problem by a…

In this paper we establish a connection between non-convex optimization methods for training deep neural networks and nonlinear partial differential equations (PDEs). Relaxation techniques arising in statistical physics which have already…

High-dimensional PDEs have been a longstanding computational challenge. We propose to solve high-dimensional PDEs by approximating the solution with a deep neural network which is trained to satisfy the differential operator, initial…

We consider the time discretization of a linear parabolic problem by the discontinuous Galerkin (DG) method using piecewise polynomials of degree at most $r-1$ in $t$, for $r\ge1$ and with maximum step size~$k$. It is well known that the…

We propose a very general framework for deriving rigorous bounds on the approximation error for physics-informed neural networks (PINNs) and operator learning architectures such as DeepONets and FNOs as well as for physics-informed operator…

We propose a neural network-based algorithm for solving forward and inverse problems for partial differential equations in unsupervised fashion. The solution is approximated by a deep neural network which is the minimizer of a cost…

We extend the Deep Galerkin Method (DGM) introduced in Sirignano and Spiliopoulos (2018)} to solve a number of partial differential equations (PDEs) that arise in the context of optimal stochastic control and mean field games. First, we…

Numerical simulations of physical systems exhibit discrepancies arising from unmodeled physics and idealizations, as well as numerical approximation errors stemming from discretization and solver tolerances. This article reviews techniques…

The solution to partial differential equations using deep learning approaches has shown promising results for several classes of initial and boundary-value problems. However, their ability to surpass, particularly in terms of accuracy,…

Recently, artificial neural networks (ANNs) in conjunction with stochastic gradient descent optimization methods have been employed to approximately compute solutions of possibly rather high-dimensional partial differential equations…

The multiscale complexity of modern problems in computational science and engineering can prohibit the use of traditional numerical methods in multi-dimensional simulations. Therefore, novel algorithms are required in these situations to…

Previous research on PAC-Bayes learning theory has focused extensively on establishing tight upper bounds for test errors. A recently proposed training procedure called PAC-Bayes training, updates the model toward minimizing these bounds.…

In recent years, there has been an increasing interest in using deep learning and neural networks to tackle scientific problems, particularly in solving partial differential equations (PDEs). However, many neural network-based methods, such…

Recently, various evolutionary partial differential equations (PDEs) with a mixed derivative have been emerged and drawn much attention. Nonetheless, their PDE-theoretical and numerical studies are still in their early stage. In this paper,…

We consider recent work of Haber and Ruthotto 2017 and Chang et al. 2018, where deep learning neural networks have been interpreted as discretisations of an optimal control problem subject to an ordinary differential equation constraint. We…