Related papers: Quantitative Approximation for Neural Operators in…
This work focuses on developing methods for approximating the solution operators of a class of parametric partial differential equations via neural operators. Neural operators have several challenges, including the issue of generating…
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…
Numerical approximations of partial differential equations (PDEs) are routinely employed to formulate the solution of physics, engineering, and mathematical problems involving functions of several variables, such as the propagation of heat…
Neural operators have emerged as a powerful tool for learning mappings between infinite-dimensional function spaces. However, their approximation properties in Sobolev norms remain poorly quantified, even though these norms control both…
We prove error bounds for operator surrogates of solution operators for partial differential and boundary integral equations on families of domains which are diffeomorphic to one common reference (or latent) domain $D_{ref}$. The pullback…
Neural operator learning models have emerged as very effective surrogates in data-driven methods for partial differential equations (PDEs) across different applications from computational science and engineering. Such operator learning…
Neural operators have emerged as transformative tools for learning mappings between infinite-dimensional function spaces, offering useful applications in solving complex partial differential equations (PDEs). This paper presents a rigorous…
The Monte Carlo-type Neural Operator (MCNO) introduces a lightweight architecture for learning solution operators for parametric PDEs by directly approximating the kernel integral using a Monte Carlo approach. Unlike Fourier Neural…
Simulations of the dynamics generated by partial differential equations (PDEs) provide approximate, numerical solutions to initial value problems. Such simulations are ubiquitous in scientific computing, but the correctness of the results…
Recent advances in scientific machine learning (SciML) have enabled neural operators (NOs) to serve as powerful surrogates for modeling the dynamic evolution of physical systems governed by partial differential equations (PDEs). While…
Fast and accurate solutions of time-dependent partial differential equations (PDEs) are of pivotal interest to many research fields, including physics, engineering, and biology. Generally, implicit/semi-implicit schemes are preferred over…
Partial differential equations (PDEs) are central to modeling physical and engineering systems, but repeatedly solving parametric PDEs remains computationally expensive. Operator learning enables fast surrogate inference, yet typically…
We study the Cauchy problem for fully nonlinear (stochastic) parabolic partial differential equations. We provide both in deterministic and stochastic case the existence of a maximal defined solution for the problem and we provide suitable…
We investigate the potential of applying (D)NN ((deep) neural networks) for approximating nonlinear mappings arising in the finite element discretization of nonlinear PDEs (partial differential equations). As an application, we apply the…
We introduce the neural tangent kernel (NTK) regime for two-layer neural operators and analyze their generalization properties. For early-stopped gradient descent (GD), we derive fast convergence rates that are known to be minimax optimal…
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…
The computational overhead of traditional numerical solvers for partial differential equations (PDEs) remains a critical bottleneck for large-scale parametric studies and design optimization. We introduce a Minimal-Data Parametric Neural…
The full history recursive multilevel Picard approximation method for semilinear parabolic partial differential equations (PDEs) is the only method which provably overcomes the curse of dimensionality for general time horizons if the…
We present a lightweighted neural PDE representation to discover the hidden structure and predict the solution of different nonlinear PDEs. Our key idea is to leverage the prior of ``translational similarity'' of numerical PDE differential…
An explicit formula to find symmetry recursion operators for partial differential equations (PDEs) is obtained from new results connecting variational integrating factors and non-variational symmetries. The formula is special case of a…