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Surrogate modeling of costly mathematical models representing physical systems is challenging since it is typically not possible to create a large experimental design. Thus, it is beneficial to constrain the approximation to adhere to the…
To solve high-dimensional parameter-dependent partial differential equations (pPDEs), a neural network architecture is presented. It is constructed to map parameters of the model data to corresponding finite element solutions. To improve…
Stochastic differential equations (SDEs) and the Kolmogorov partial differential equations (PDEs) associated to them have been widely used in models from engineering, finance, and the natural sciences. In particular, SDEs and Kolmogorov…
Partial differential equations (PDEs) govern a wide range of physical phenomena, but their numerical solution remains computationally demanding, especially when repeated simulations are required across many parameter settings. Recent…
Estimating matrices in the symmetric positive-definite (SPD) cone is of interest for many applications ranging from computer vision to graph learning. While there exist various convex optimization-based estimators, they remain limited in…
Neural operators have been validated as promising deep surrogate models for solving partial differential equations (PDEs). Despite the critical role of boundary conditions in PDEs, however, only a limited number of neural operators robustly…
Traffic state estimation (TSE) fundamentally involves solving high-dimensional spatiotemporal partial differential equations (PDEs) governing traffic flow dynamics from limited, noisy measurements. While Physics-Informed Neural Networks…
Partial differential equations (PDEs) underlie our understanding and prediction of natural phenomena across numerous fields, including physics, engineering, and finance. However, solving parametric PDEs is a complex task that necessitates…
Discrete Differential Equations (DDEs) are functional equations that relate polynomially a power series $F(t,u)$ in $t$ with polynomial coefficients in a "catalytic" variable $u$ and the specializations, say at $u=1$, of $F(t,u)$ and of…
While it is widely known that neural networks are universal approximators of continuous functions, a less known and perhaps more powerful result is that a neural network with a single hidden layer can approximate accurately any nonlinear…
In the present work, a multi-scale framework for neural network enhanced methods is proposed for approximation of function and solution of partial differential equations (PDEs). By introducing the multi-scale concept, the total solution of…
The traditional limitations of neural networks in reliably generalizing beyond the convex hulls of their training data present a significant problem for computational physics, in which one often wishes to solve PDEs in regimes far beyond…
Deep neural networks are powerful learning models that achieve state-of-the-art performance on many computer vision, speech, and language processing tasks. In this paper, we study a fundamental question that arises when designing deep…
Deep neural operators can learn nonlinear mappings between infinite-dimensional function spaces via deep neural networks. As promising surrogate solvers of partial differential equations (PDEs) for real-time prediction, deep neural…
Partial differential equations (PDEs) play a fundamental role in modeling and simulating problems across a wide range of disciplines. Recent advances in deep learning have shown the great potential of physics-informed neural networks…
In numerous contexts, high-resolution solutions to partial differential equations are required to capture faithfully essential dynamics which occur at small spatiotemporal scales, but these solutions can be very difficult and slow to obtain…
A key challenge in scientific machine learning is solving partial differential equations (PDEs) on complex domains, where the curved geometry complicates the approximation of functions and their derivatives required by differential…
Unlike classical artificial neural networks, which require retraining for each new set of parametric inputs, the Deep Operator Network (DeepONet), a lately introduced deep learning framework, approximates linear and nonlinear solution…
Deep operator network (DeepONet) has shown significant promise as surrogate models for systems governed by partial differential equations (PDEs), enabling accurate mappings between infinite-dimensional function spaces. However, when applied…
We introduce a novel neural operator architecture designed to approximate solutions of linear elliptic partial differential equations with high-contrast, spatially varying coefficients. The network, termed the Iterated V-shaped Net…