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A new data-driven method for operator learning of stochastic differential equations(SDE) is proposed in this paper. The central goal is to solve forward and inverse stochastic problems more effectively using limited data. Deep operator…

Data-driven machine learning approaches are being increasingly used to solve partial differential equations (PDEs). They have shown particularly striking successes when training an operator, which takes as input a PDE in some family, and…

Neural ordinary differential equations (NODEs) presented a new paradigm to construct (continuous-time) neural networks. While showing several good characteristics in terms of the number of parameters and the flexibility in constructing…

Neural operators are a type of deep architecture that learns to solve (i.e. learns the nonlinear solution operator of) partial differential equations (PDEs). The current state of the art for these models does not provide explicit…

Existing work in scientific machine learning (SciML) has shown that data-driven learning of solution operators can provide a fast approximate alternative to classical numerical partial differential equation (PDE) solvers. Of these, Neural…

Neural operators have become increasingly popular in solving \textit{partial differential equations} (PDEs) due to their superior capability to capture intricate mappings between function spaces over complex domains. However, the…

Learning underlying dynamics from data is important and challenging in many real-world scenarios. Incorporating differential equations (DEs) to design continuous networks has drawn much attention recently, however, most prior works make…

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…

The neural operator has emerged as a powerful tool in learning mappings between function spaces in PDEs. However, when faced with real-world physical data, which are often highly non-uniformly distributed, it is challenging to use…

Many important problems in science and engineering require solving the so-called parametric partial differential equations (PDEs), i.e., PDEs with different physical parameters, boundary conditions, shapes of computational domains, etc.…

Numerous physics theories are rooted in partial differential equations (PDEs). However, the increasingly intricate physics equations, especially those that lack analytic solutions or closed forms, have impeded the further development of…

Modeling nonlinear spatiotemporal dynamical systems has primarily relied on partial differential equations (PDEs). However, the explicit formulation of PDEs for many underexplored processes, such as climate systems, biochemical reaction and…

Recently deep learning surrogates and neural operators have shown promise in solving partial differential equations (PDEs). However, they often require a large amount of training data and are limited to bounded domains. In this work, we…

Single-operator learning involves training a deep neural network to learn a specific operator, whereas recent work in multi-operator learning uses an operator embedding structure to train a single neural network on data from multiple…

Stochastic differential equations (SDEs) and stochastic partial differential equations (SPDEs) are fundamental for modeling stochastic dynamics across the natural sciences and modern machine learning. Learning their solution operators with…

Neural operators have emerged as powerful deep learning frameworks for approximating solution operators of parameterized partial differential equations (PDE). However, current methods predominantly rely on multilayer perceptrons (MLPs) for…

This work proposes an autoencoder neural network as a non-linear generalization of projection-based methods for solving Partial Differential Equations (PDEs). The proposed deep learning architecture presented is capable of generating the…

The predictive accuracy of operator learning frameworks depends on the quality and quantity of available training data (input-output function pairs), often requiring substantial amounts of high-fidelity data, which can be challenging to…

Neural operators have emerged as a powerful tool for learning the mapping between infinite-dimensional parameter and solution spaces of partial differential equations (PDEs). In this work, we focus on multiscale PDEs that have important…

The finite element method (FEM) is a well-established numerical method for solving partial differential equations (PDEs). However, its mesh-based nature gives rise to substantial computational costs, especially for complex multiscale…