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Simulating reactive dissolution of solid minerals in porous media has many subsurface applications, including carbon capture and storage (CCS), geothermal systems and oil & gas recovery. As traditional direct numerical simulators are…
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…
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…
In X-ray Computed Tomography (CT), projections from many angles are acquired and used for 3D reconstruction. To make CT suitable for in-line quality control, reducing the number of angles while maintaining reconstruction quality is…
Physics-informed neural networks (PINNs) effectively embed physical principles into machine learning, but often struggle with complex or alternating geometries. We propose a novel method for integrating geometric transformations within…
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…
Finding accurate solutions to partial differential equations (PDEs) is a crucial task in all scientific and engineering disciplines. It has recently been shown that machine learning methods can improve the solution accuracy by correcting…
Recently, researchers have utilized neural networks to accurately solve partial differential equations (PDEs), enabling the mesh-free method for scientific computation. Unfortunately, the network performance drops when encountering a high…
We present a framework for solving time-dependent partial differential equations (PDEs) in the spirit of the random feature method. The numerical solution is constructed using a space-time partition of unity and random feature functions.…
In this paper, based on the idea of direct discrete modeling (DDM) with equilibrium distribution functions (EDFs), we develop a general framework of the mesoscopic numerical method (MesoNM) for macroscopic partial differential equations…
Partial differential equations (PDEs) are crucial for modeling various physical phenomena such as heat transfer, fluid flow, and electromagnetic waves. In computer-aided engineering (CAE), the ability to handle fine resolutions and large…
Transfer learning for partial differential equations (PDEs) is to develop a pre-trained neural network that can be used to solve a wide class of PDEs. Existing transfer learning approaches require much information of the target PDEs such as…
Deep learning methods based on backward stochastic differential equations (BSDEs) have emerged as competitive alternatives to physics-informed neural networks (PINNs) for solving high-dimensional partial differential equations (PDEs). By…
We investigate a deep learning approach to efficiently perform Bayesian inference in partial differential equation (PDE) and integral equation models over potentially high-dimensional parameter spaces. The contributions of this paper are…
Developing fast surrogates for Partial Differential Equations (PDEs) will accelerate design and optimization in almost all scientific and engineering applications. Neural networks have been receiving ever-increasing attention and…
Spatial prediction refers to the estimation of unobserved values from spatially distributed observations. Although recent advances have improved the capacity to model diverse observation types, adoption in practice remains limited in…
A statistical learning approach for parametric PDEs related to Uncertainty Quantification is derived. The method is based on the minimization of an empirical risk on a selected model class and it is shown to be applicable to a broad range…
The numerical solution of high dimensional partial differential equations (PDEs) is severely constrained by the curse of dimensionality (CoD), rendering classical grid--based methods impractical beyond a few dimensions. In recent years,…
Critical aspects of computational imaging systems, such as experimental design and image priors, can be optimized through deep networks formed by the unrolled iterations of classical model-based reconstructions (termed physics-based…
Deep neural networks (DNNs), especially physics-informed neural networks (PINNs), have recently become a new popular method for solving forward and inverse problems governed by partial differential equations (PDEs). However, these methods…