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This article introduces GIT-Net, a deep neural network architecture for approximating Partial Differential Equation (PDE) operators, inspired by integral transform operators. GIT-NET harnesses the fact that differential operators commonly…

Uncertainty quantification is a fundamental yet unsolved problem for deep learning. The Bayesian framework provides a principled way of uncertainty estimation but is often not scalable to modern deep neural nets (DNNs) that have a large…

We construct and analyze approximation rates of deep operator networks (ONets) between infinite-dimensional spaces that emulate with an exponential rate of convergence the coefficient-to-solution map of elliptic second-order partial…

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…

In this article, we investigate the existence of a deep neural network (DNN) capable of approximating solutions to partial integro-differential equations while circumventing the curse of dimensionality. Using the Feynman-Kac theorem, we…

Deep Operator Networks (DeepOnets) have revolutionized the domain of scientific machine learning for the solution of the inverse problem for dynamical systems. However, their implementation necessitates optimizing a high-dimensional space…

Recent experiments have shown that deep networks can approximate solutions to high-dimensional PDEs, seemingly escaping the curse of dimensionality. However, questions regarding the theoretical basis for such approximations, including the…

In this paper, we study a machine-learning-based solver for high-dimensional partial differential equations (PDEs). Computing accurate solutions efficiently for such problems remains challenging because of the curse of dimensionality, which…

Deep neural operators are recognized as an effective tool for learning solution operators of complex partial differential equations (PDEs). As compared to laborious analytical and computational tools, a single neural operator can predict…

In many mechanistic medical, biological, physical and engineered spatiotemporal dynamic models the numerical solution of partial differential equations (PDEs) can make simulations impractically slow. Biological models require the…

A state-of-the-art deep domain decomposition method (D3M) based on the variational principle is proposed for partial differential equations (PDEs). The solution of PDEs can be formulated as the solution of a constrained optimization…

Polynomial chaos expansions (PCE) are widely used in the framework of uncertainty quantification. However, when dealing with high dimensional complex problems, challenging issues need to be faced. For instance, high-order polynomials may be…

Transfer learning for partial differential equations (PDEs) is to develop a pre-trained neural network that can be used to solve a wide class of PDEs. Existing transfer learning approaches require much information of the target PDEs such as…

We present a data-driven control framework for partial differential equations (PDEs). Our approach integrates Time-Integrated Deep Operator Networks (TI-DeepONets) as differentiable PDE surrogate models within the Differentiable Predictive…

We consider using deep neural networks to solve time-dependent partial differential equations (PDEs), where multi-scale processing is crucial for modeling complex, time-evolving dynamics. While the U-Net architecture with skip connections…

Model discrepancy, defined as the difference between model predictions and reality, is ubiquitous in computational models for physical systems. It is common to derive partial differential equations (PDEs) from first principles physics, but…

It is one of the most challenging issues in applied mathematics to approximately solve high-dimensional partial differential equations (PDEs) and most of the numerical approximation methods for PDEs in the scientific literature suffer from…

In this paper we propose a new model-based unsupervised learning method, called VarNet, for the solution of partial differential equations (PDEs) using deep neural networks (NNs). Particularly, we propose a novel loss function that relies…

As supported by abundant experimental evidence, neural networks are state-of-the-art for many approximation tasks in high-dimensional spaces. Still, there is a lack of a rigorous theoretical understanding of what they can approximate, at…

Neural PDE solvers are increasingly used as learned surrogates for families of partial differential equations, where the key machine learning challenge is not only interpolation on a fixed benchmark distribution but generalization under…