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Neural operators have shown promise in solving many types of Partial Differential Equations (PDEs). They are significantly faster compared to traditional numerical solvers once they have been trained with a certain amount of observed data.…
Using the general formalism of [12], a study of index theory for non-Fredholm operators was initiated in [9]. Natural examples arise from $(1+1)$-dimensional differential operators using the model operator $D_A$ in $L^2(\mathbb{R}^2; dt…
The Physics-Informed Neural Network (PINN) framework introduced recently incorporates physics into deep learning, and offers a promising avenue for the solution of partial differential equations (PDEs) as well as identification of the…
Neural operators have emerged as fast surrogate solvers for parametric partial differential equations (PDEs). However, purely data-driven models often require extensive training data and can generalize poorly, especially in small-data…
We propose a novel machine learning framework for solving optimization problems governed by large-scale partial differential equations (PDEs) with high-dimensional random parameters. Such optimization under uncertainty (OUU) problems may be…
Solving nonlinear partial differential equations (PDEs) with multiple solutions using neural networks has found widespread applications in various fields such as physics, biology, and engineering. However, classical neural network methods…
We study Fredholm determinants of a class of integral operators, whose kernels can be expressed as double contour integrals of a special type. Such Fredholm determinants appear in various random matrix and statistical physics models. We…
Accurately quantifying long-term risk probabilities in diverse stochastic systems is essential for safety-critical control. However, existing sampling-based and partial differential equation (PDE)-based methods often struggle to handle…
Coupled partial differential equations (PDEs) are key tasks in modeling the complex dynamics of many physical processes. Recently, neural operators have shown the ability to solve PDEs by learning the integral kernel directly in…
Motivated by the dynamics of defects in planar pattern-forming systems, we study Fredholm properties of elliptic operators with singular coefficients in weighted Sobolev spaces. In particular, we consider a family of doubly weighted spaces…
We propose the first learning scheme for functional differential equations (FDEs). FDEs play a fundamental role in physics, mathematics, and optimal control. However, the numerical analysis of FDEs has faced challenges due to its…
Recent advancements in operator-type neural networks have shown promising results in approximating the solutions of spatiotemporal Partial Differential Equations (PDEs). However, these neural networks often entail considerable training…
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…
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…
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…
A kernel-based approach for the learning of the solution operator of general nonhomogeneous partial differential equations (PDEs) is proposed. The method incorporates physical priors, typically encoded through the PDE operator, into a…
In scientific and engineering applications, solving partial differential equations (PDEs) across various parameters and domains normally relies on resource-intensive numerical methods. Neural operators based on deep learning offered a…
Neural operators have emerged as a promising paradigm for learning solution operators of partial differential equa- tions (PDEs) directly from data. Existing methods, such as those based on Fourier or graph techniques, make strong as-…
Cellular metamaterials offer a vast design space for tailoring nonlinear mechanical responses, yet exploring this space with conventional modeling approaches is often infeasible or not scalable. To fully exploit their nonlinear behavior for…
Fourier neural operators (FNOs) can learn highly nonlinear mappings between function spaces, and have recently become a popular tool for learning responses of complex physical systems. However, to achieve good accuracy and efficiency, FNOs…