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Modeling high-frequency information is a critical challenge in scientific machine learning. For instance, fully turbulent flow simulations of the Navier-Stokes equations at Reynolds numbers 3500 and above can generate high-frequency signals…

The scientific computation methods development in conjunction with artificial intelligence technologies remains a hot research topic. Finding a balance between lightweight and accurate computations is a solid foundation for this direction.…

We study learning weak solutions to nonlinear hyperbolic partial differential equations (H-PDE), which have been difficult to learn due to discontinuities in their solutions. We use a physics-informed variant of the Fourier Neural Operator…

Optimal control problems with nonsmooth objectives and nonlinear partial differential equation (PDE) constraints are challenging, mainly because of the underlying nonsmooth and nonconvex structures and the demanding computational cost for…

Recently, operator learning, or learning mappings between infinite-dimensional function spaces, has garnered significant attention, notably in relation to learning partial differential equations from data. Conceptually clear when outlined…

Integrating machine learning techniques in established numerical solvers represents a modern approach to enhancing computational fluid dynamics simulations. Within the lattice Boltzmann method (LBM), the collision operator serves as an…

Neural operators have achieved strong performance in learning solution operators of partial differential equations (PDEs), but their inherently continuous representations struggle to capture discontinuities and sharp transitions. Existing…

This paper introduces significant advancements in fractional neural operators (FNOs) through the integration of adaptive hybrid kernels and stochastic multiscale analysis. We address several open problems in the existing literature by…

Memory complexity and data scarcity have so far prohibited learning solution operators of partial differential equations (PDEs) at high resolutions. We address these limitations by introducing a new data efficient and highly parallelizable…

Magnetotelluric (MT) forward modeling is fundamental for improving the accuracy and efficiency of MT inversion. Neural operators (NOs) have been effectively used for rapid MT forward modeling, demonstrating their promising performance in…

Neural operators extend data-driven models to map between infinite-dimensional functional spaces. These models have successfully solved continuous dynamical systems represented by differential equations, viz weather forecasting, fluid flow,…

We show that the error achievable using physics-informed neural networks for solving systems of differential equations can be substantially reduced when these networks are trained using meta-learned optimization methods rather than to using…

We propose a trainable-by-parts surrogate model for solving forward and inverse parameterized nonlinear partial differential equations. Like several other surrogate and operator learning models, the proposed approach employs an encoder to…

In this work, we propose a concise neural operator architecture for operator learning. Drawing an analogy with a conventional fully connected neural network, we define the neural operator as follows: the output of the $i$-th neuron in a…

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…

Neural operators approximate PDE solution maps, but they need not respect the symmetries of the governing equation. In out-of-distribution (OOD) regimes, a standard neural operator must often learn coordinate alignment and physical…

This article introduces a novel approach to learning monotone neural networks through a newly defined penalization loss. The proposed method is particularly effective in solving classes of variational problems, specifically monotone…

Neural ordinary differential equations (NODEs) -- parametrizations of differential equations using neural networks -- have shown tremendous promise in learning models of unknown continuous-time dynamical systems from data. However, every…

Physics-Informed Neural Networks (PINNs) and Neural Ordinary Differential Equations (NODEs) represent two distinct machine learning frameworks for modeling nonlinear neuronal dynamics. This study systematically evaluates their performance…

In this paper, we propose using LSTM-RNNs (Long Short-Term Memory-Recurrent Neural Networks) to learn and represent nonlinear integral operators that appear in nonlinear integro-differential equations (IDEs). The LSTM-RNN representation of…