Related papers: Quantitative Approximation for Neural Operators in…
We develop a rigorous framework for extending neural operators to handle out-of-distribution input functions. We leverage kernel approximation techniques and provide theory for characterizing the input-output function spaces in terms of…
We develop a novel computational framework to approximate solution operators of evolution partial differential equations (PDEs). By employing a general nonlinear reduced-order model, such as a deep neural network, to approximate the…
In this paper, we considered the problem of the simultaneous approximation of a function and its derivatives by means of the well-known neural network (NN) operators activated by sigmoidal function. Other than a uniform convergence theorem…
We derive upper bounds on the complexity of ReLU neural networks approximating the solution maps of parametric partial differential equations. In particular, without any knowledge of its concrete shape, we use the inherent…
Kernel-based approach to operator approximation for partial differential equations has been shown to be unconditionally stable for linear PDEs and numerically exhibit unconditional stability for non-linear PDEs. These methods have the same…
The classical development of neural networks has primarily focused on learning mappings between finite-dimensional Euclidean spaces. Recently, this has been generalized to neural operators that learn mappings between function spaces. For…
Partial differential equations (PDEs) underpin the modeling of many natural and engineered systems. It can be convenient to express such models as neural PDEs rather than using traditional numerical PDE solvers by replacing part or all of…
We develop a framework for estimating unknown partial differential equations from noisy data, using a deep learning approach. Given noisy samples of a solution to an unknown PDE, our method interpolates the samples using a neural network,…
Pretraining for partial differential equation (PDE) modeling has recently shown promise in scaling neural operators across datasets to improve generalizability and performance. Despite these advances, our understanding of how pretraining…
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…
Deep neural networks (DNNs) have recently emerged as effective tools for approximating solution operators of partial differential equations (PDEs) including evolutionary problems. Classical numerical solvers for such PDEs often face…
A large class of hyperbolic and advection-dominated PDEs can have solutions with discontinuities. This paper investigates, both theoretically and empirically, the operator learning of PDEs with discontinuous solutions. We rigorously prove,…
In this paper we introduce a multilevel Picard approximation algorithm for general semilinear parabolic PDEs with gradient-dependent nonlinearities whose coefficient functions do not need to be constant. We also provide a full convergence…
Can neural networks learn to solve partial differential equations (PDEs)? We investigate this question for two (systems of) PDEs, namely, the Poisson equation and the steady Navier--Stokes equations. The contributions of this paper are…
In a recent article, we presented a framework to control nonlinear partial differential equations (PDEs) by means of Koopman operator based reduced models and concepts from switched systems. The main idea was to transform a control system…
Neural operators are a popular technique in scientific machine learning to learn a mathematical model of the behavior of unknown physical systems from data. Neural operators are especially useful to learn solution operators associated with…
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…
Recently, deep Convolutional Neural Networks (CNNs) have proven to be successful when employed in areas such as reduced order modeling of parametrized PDEs. Despite their accuracy and efficiency, the approaches available in the literature…
Fractional differential operators provide an attractive mathematical tool to model effects with limited regularity properties. Particular examples are image processing and phase field models in which jumps across lower dimensional subsets…
Fourier Neural Operators are deep learning models that learn mappings between function spaces and can be used to learn and solve partial differential equations (PDEs), in some cases significantly faster than traditional PDE solvers. Within…