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The reconstruction and inference of stochastic dynamical systems from data is a fundamental task in inverse problems and statistical learning. While surrogate modeling advances computational methods to approximate these dynamics, standard…

Inverse problems in partial differential equations (PDEs) involve estimating the physical parameters of a system from observed spatiotemporal solution fields. Neural networks are well-suited for PDE parameter estimation due to their…

Partial differential equations (PDEs) are commonly derived based on empirical observations. However, recent advances of technology enable us to collect and store massive amount of data, which offers new opportunities for data-driven…

Neural networks are one tool for approximating non-linear differential equations used in scientific computing tasks such as surrogate modeling, real-time predictions, and optimal control. PDE foundation models utilize neural networks to…

Solving analytically intractable partial differential equations (PDEs) that involve at least one variable defined on an unbounded domain arises in numerous physical applications. Accurately solving unbounded domain PDEs requires efficient…

This work formulates a new approach to reduced modeling of parameterized, time-dependent partial differential equations (PDEs). The method employs Operator Inference, a scientific machine learning framework combining data-driven learning…

We introduce a compositional physics-aware FInite volume Neural Network (FINN) for learning spatiotemporal advection-diffusion processes. FINN implements a new way of combining the learning abilities of artificial neural networks with…

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…

Solving time-dependent partial differential equations (PDEs) that exhibit sharp gradients or local singularities is computationally demanding, as traditional physics-informed neural networks (PINNs) often suffer from inefficient point…

Spectral methods are an important part of scientific computing's arsenal for solving partial differential equations (PDEs). However, their applicability and effectiveness depend crucially on the choice of basis functions used to expand the…

Partial differential equations (PDEs) are central to scientific modeling. Modern workflows increasingly rely on learning-based components to support model reuse, inference, and integration across large computational processes. Despite the…

Ordinary and partial differential equations (ODEs/PDEs) play a paramount role in analyzing and simulating complex dynamic processes across all corners of science and engineering. In recent years machine learning tools are aspiring to…

Physics-informed deep operator networks (DeepONets) have emerged as a promising approach toward numerically approximating the solution of partial differential equations (PDEs). In this work, we aim to develop further understanding of what…

We present a framework and algorithms to learn controlled dynamics models using neural stochastic differential equations (SDEs) -- SDEs whose drift and diffusion terms are both parametrized by neural networks. We construct the drift term to…

The fractional advection-dispersion equation (FADE) has attracted increased attention from researchers as it provides an accurate description for challenging phenomenas with long-range time memory and spatial interactions, such as the…

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…

Many scientific phenomena are modeled by Partial Differential Equations (PDEs). The development of data gathering tools along with the advances in machine learning (ML) techniques have raised opportunities for data-driven identification of…

We introduce Neural Parameter Regression (NPR), a novel framework specifically developed for learning solution operators in Partial Differential Equations (PDEs). Tailored for operator learning, this approach surpasses traditional DeepONets…

I provide an introduction to the application of deep learning and neural networks for solving partial differential equations (PDEs). The approach, known as physics-informed neural networks (PINNs), involves minimizing the residual of the…

We present a data-driven control framework for partial differential equations (PDEs). Our approach integrates Time-Integrated Deep Operator Networks (TI-DeepONets) as differentiable PDE surrogate models within the Differentiable Predictive…