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Recent studies have demonstrated the success of deep learning in solving forward and inverse problems in engineering and scientific computing domains, such as physics-informed neural networks (PINNs). Source inversion problems under sparse…
This work is concerned with discovering the governing partial differential equation (PDE) of a physical system. Existing methods have demonstrated the PDE identification from finite observations but failed to maintain satisfying results…
The recently introduced Physics-Informed Neural Networks (PINNs) have popularized least squares formulations of both forward and inverse problems involving partial differential equations (PDEs) in strong form. We employ both Isogeometric…
Recent years have witnessed a growth in mathematics for deep learning--which seeks a deeper understanding of the concepts of deep learning with mathematics and explores how to make it more robust--and deep learning for mathematics, where…
Modeling stiff partial differential equations (PDEs) with sharp gradients remains a significant challenge for scientific machine learning. While Physics-Informed Neural Networks (PINNs) struggle with spectral bias and slow training times,…
Identifying parameters in partial differential equations (PDEs) represents a very broad class of applied inverse problems. In recent years, several unsupervised learning approaches using (deep) neural networks have been developed to solve…
Partial differential equation (PDE) solvers underpin modern quantitative finance, governing option pricing and risk evaluation. Physics-Informed Neural Networks (PINNs) have emerged as a promising approach for solving the forward and…
In recent years, there has been a growing interest in leveraging deep learning and neural networks to address scientific problems, particularly in solving partial differential equations (PDEs). However, many neural network-based methods…
Physics Informed Neural Networks (PINNs) have frequently been used for the numerical approximation of Partial Differential Equations (PDEs). The goal of this paper is to construct PINNs along with a computable upper bound of the error,…
We propose a methodology that combines generative latent diffusion models with physics-informed machine learning to generate solutions of parametric partial differential equations (PDEs) conditioned on partial observations, which includes,…
In this paper, we address the issue of model specification in probabilistic latent variable models (PLVMs) using an infinite-horizon optimal control approach. Traditional PLVMs rely on joint distributions to model complex data, but…
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…
In this paper, we introduce a novel, data-driven approach for solving high-dimensional Bayesian inverse problems based on partial differential equations (PDEs), called Weak Neural Variational Inference (WNVI). The method complements real…
A state-of-the-art deep domain decomposition method (D3M) based on the variational principle is proposed for partial differential equations (PDEs). The solution of PDEs can be formulated as the solution of a constrained optimization…
Solving high-dimensional parabolic partial differential equations (PDEs) with deep learning methods is often computationally and memory intensive, primarily due to the need for automatic differentiation (AD) to compute large Hessian…
Partial differential equations (PDEs) on surfaces are fundamental to scientific computing and geometry processing. A popular approach to solving PDEs on surfaces is the finite element method (FEM), where the surface is divided into discrete…
Deep latent variable models (DLVMs) combine the approximation abilities of deep neural networks and the statistical foundations of generative models. Variational methods are commonly used for inference; however, the exact likelihood of…
State-dependent parameter identification, where unknown model parameters depend on one or more state variables in partial differential equations (PDEs) or coupled PDE systems, is fundamental to a wide range of problems in physics,…
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…
Parameter estimation for differential equations from measured data is an inverse problem prevalent across quantitative sciences. Physics-Informed Neural Networks (PINNs) have emerged as effective tools for solving such problems, especially…