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Partial differential equations often contain unknown functions that are difficult or impossible to measure directly, hampering our ability to derive predictions from the model. Workflows for recovering scalar PDE parameters from data are…

Partial differential equations (PDEs) involving high contrast and oscillating coefficients are common in scientific and industrial applications. Numerical approximation of these PDEs is a challenging task that can be addressed, for example,…

A novel multi-level method for partial differential equations with uncertain parameters is proposed. The principle behind the method is that the error between grid levels in multi-level methods has a spatial structure that is by good…

Traditional data-driven deep learning models often struggle with high training costs, error accumulation, and poor generalizability in complex physical processes. Physics-informed deep learning (PiDL) addresses these challenges by…

In recent years, deep perceptual loss has been widely and successfully used to train machine learning models for many computer vision tasks, including image synthesis, segmentation, and autoencoding. Deep perceptual loss is a type of loss…

Loss landscape is a useful tool to characterize and compare neural network models. The main challenge for analysis of loss landscape for the deep neural networks is that they are generally highly non-convex in very high dimensional space.…

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…

Neural networks have been very successful in many applications; we often, however, lack a theoretical understanding of what the neural networks are actually learning. This problem emerges when trying to generalise to new data sets. The…

This paper presents machine learning techniques and deep reinforcement learningbased algorithms for the efficient resolution of nonlinear partial differential equations and dynamic optimization problems arising in investment decisions and…

Deep Metric Learning (DML) aims to learn embedding functions that map semantically similar inputs to proximate points in a metric space while separating dissimilar ones. Existing methods, such as pairwise losses, are hindered by complex…

In this paper, we propose a novel numerical method for Path-Dependent Partial Differential Equations (PPDEs). These equations firstly appeared in the seminal work of Dupire [2009], where the functional It\^o calculus was developed to deal…

In this article, we present a three-dimensional anisotropic $hp$-mesh refinement strategy for ultraweak discontinuous Petrov--Galerkin (DPG) formulations with optimal test functions. The refinement strategy utilizes the built-in…

Nonlinear differential equations are challenging to solve numerically and are important to understanding the dynamics of many physical systems. Deep neural networks have been applied to help alleviate the computational cost that is…

This article presents a general approach akin to domain-decomposition methods to solve a single linear PDE, but where each subdomain of a partitioned domain is associated to a distinct variational formulation coming from a mutually…

Parametric partial differential equations (PDEs) are fundamental for modeling a wide range of physical and engineering systems influenced by uncertain or varying parameters. Traditional neural network-based solvers, such as Physics-Informed…

The combination of Monte Carlo methods and deep learning has recently led to efficient algorithms for solving partial differential equations (PDEs) in high dimensions. Related learning problems are often stated as variational formulations…

Gaussian processes (GPs) are nonparametric priors over functions. Fitting a GP implies computing a posterior distribution of functions consistent with the observed data. Similarly, deep Gaussian processes (DGPs) should allow us to compute a…

Many problems in science and engineering can be represented by a set of partial differential equations (PDEs) through mathematical modeling. Mechanism-based computation following PDEs has long been an essential paradigm for studying topics…

Metric learning has become an attractive field for research on the latest years. Loss functions like contrastive loss, triplet loss or multi-class N-pair loss have made possible generating models capable of tackling complex scenarios with…

The discontinuous Petrov-Galerkin method is a minimal residual method with broken test spaces and is introduced for a nonlinear model problem in this paper. Its lowest-order version applies to a nonlinear uniformly convex model example and…