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Data-driven surrogate models are widely used for applications such as design optimization and uncertainty quantification, where repeated evaluations of an expensive simulator are required. For most partial differential equation (PDE)…

A primary challenge of physics-informed machine learning (PIML) is its generalization beyond the training domain, especially when dealing with complex physical problems represented by partial differential equations (PDEs). This paper aims…

Deep surrogate models for parametric partial differential equations (PDEs) can deliver high-fidelity approximations but remain prohibitively data-hungry: training often requires thousands of fine-grid simulations, each incurring substantial…

Deep learning has shown strong potential in modeling complex spatiotemporal dynamics. However, most existing methods depend on densely and uniformly sampled data, which is often unavailable in practice due to sensor and cost limitations. In…

Symbolic regression plays a crucial role in modern scientific research thanks to its capability of discovering concise and interpretable mathematical expressions from data. A key challenge lies in the search for parsimonious and…

Physics-Informed Neural Networks (PINNs) represent a significant advancement in Scientific Machine Learning (SciML), which integrate physical domain knowledge into an empirical loss function as soft constraints and apply existing machine…

In recent years, we have witnessed the emergence of scientific machine learning as a data-driven tool for the analysis, by means of deep-learning techniques, of data produced by computational science and engineering applications. At the…

Solving time-dependent Partial Differential Equations (PDEs) using a densely discretized spatial domain is a fundamental problem in various scientific and engineering disciplines, including modeling climate phenomena and fluid dynamics.…

This work presents a physics-informed deep learning-based super-resolution framework to enhance the spatio-temporal resolution of the solution of time-dependent partial differential equations (PDE). Prior works on deep learning-based…

This work proposes and analyzes a generalized acceleration technique for decreasing the computational complexity of using stochastic collocation (SC) methods to solve partial differential equations (PDEs) with random input data. The SC…

Partial differential equations (PDEs) govern physical phenomena across the full range of scientific scales, yet their computational solution remains one of the defining challenges of modern science. This critical review examines two mature…

We propose new machine learning schemes for solving high dimensional nonlinear partial differential equations (PDEs). Relying on the classical backward stochastic differential equation (BSDE) representation of PDEs, our algorithms estimate…

The term `surrogate modeling' in computational science and engineering refers to the development of computationally efficient approximations for expensive simulations, such as those arising from numerical solution of partial differential…

Physics-informed deep learning approaches have been developed to solve forward and inverse stochastic differential equation (SDE) problems with high-dimensional stochastic space. However, the existing deep learning models have difficulties…

We present Neural Spectral Methods, a technique to solve parametric Partial Differential Equations (PDEs), grounded in classical spectral methods. Our method uses orthogonal bases to learn PDE solutions as mappings between spectral…

This paper introduces a new approximation scheme for solving high-dimensional semilinear partial differential equations (PDEs) and backward stochastic differential equations (BSDEs). First, we decompose a target semilinear PDE (BSDE) into…

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…

Time-dependent partial differential equations (PDEs) often develop sharp fronts, localized peaks, and other moving structures that occupy only a small portion of the space--time domain but dominate the approximation error. This makes fixed…

We propose a physics-informed consistency modeling framework for solving partial differential equations (PDEs) via fast, few-step generative inference. We identify a key stability challenge in physics-constrained consistency training, where…

ADCME is a novel computational framework to solve inverse problems involving physical simulations and deep neural networks (DNNs). This paper benchmarks its capability to learn spatially-varying physical fields using DNNs. We demonstrate…