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Accuracy in neural PDE solvers often breaks down not because of limited expressivity, but due to poor optimisation caused by ill-conditioning, especially in multi-fidelity and stiff problems. We study this issue in Physics-Informed Extreme…
Deep learning algorithms have been successfully applied to numerically solve linear Kolmogorov partial differential equations~(PDEs). A recent research shows that if the initial functions are bounded, the empirical risk minimization (ERM)…
Deep neural networks have been successful in many predictive modeling tasks, such as image and language recognition, where large neural networks are often used to obtain good accuracy. Consequently, it is challenging to deploy these…
In this paper, we consider approximating the parameter-to-solution maps of parametric partial differential equations (PPDEs) using deep neural networks (DNNs). We propose an efficient approach combining reduced collocation methods (RCMs)…
The finite element method, finite difference method, finite volume method and spectral method have achieved great success in solving partial differential equations. However, the high accuracy of traditional numerical methods is at the cost…
In this article we present new results on neural networks with linear threshold activation functions. We precisely characterize the class of functions that are representable by such neural networks and show that 2 hidden layers are…
This work extends the paradigm of evolutional deep neural networks (EDNNs) to solving parametric time-dependent partial differential equations (PDEs) on domains with geometric structure. By introducing positional embeddings based on…
We present a novel Galerkin method for solving partial differential equations on the sphere. The problem is discretized by a highly localized basis which is easily constructed. The stiffness matrix entries are computed by a recently…
We study the Evolutionary Deep Neural Network (EDNN) framework for accelerating numerical solvers of time-dependent partial differential equations (PDEs). We introduce a Low-Rank Evolutionary Deep Neural Network (LR-EDNN), which constrains…
Within the framework of parameter dependent PDEs, we develop a constructive approach based on Deep Neural Networks for the efficient approximation of the parameter-to-solution map. The research is motivated by the limitations and drawbacks…
Recent developments in mechanical, aerospace, and structural engineering have driven a growing need for efficient ways to model and analyse structures at much larger and more complex scales than before. While established numerical methods…
A burgeoning line of research leverages deep neural networks to approximate the solutions to high dimensional PDEs, opening lines of theoretical inquiry focused on explaining how it is that these models appear to evade the curse of…
We introduce a conceptual framework for numerically solving linear elliptic, parabolic, and hyperbolic PDEs on bounded, polytopal domains in euclidean spaces by deep neural networks. The PDEs are recast as minimization of a least-squares…
We present a multidimensional deep learning implementation of a stochastic branching algorithm for the numerical solution of fully nonlinear PDEs. This approach is designed to tackle functional nonlinearities involving gradient terms of any…
We introduce a new class of spatially stochastic physics and data informed deep latent models for parametric partial differential equations (PDEs) which operate through scalable variational neural processes. We achieve this by assigning…
Partial differential equations (PDEs) are typically used as models of physical processes but are also of great interest in PDE-based image processing. However, when it comes to their use in imaging, conventional numerical methods for…
Deep neural networks (DNNs), particularly those using Rectified Linear Unit (ReLU) activation functions, have achieved remarkable success across diverse machine learning tasks, including image recognition, audio processing, and language…
Today, it is more important than ever before for users to have trust in the models they use. As Machine Learning models fall under increased regulatory scrutiny and begin to see more applications in high-stakes situations, it becomes…
Machine learning is a rapidly advancing field with diverse applications across various domains. One prominent area of research is the utilization of deep learning techniques for solving partial differential equations(PDEs). In this work, we…
One of the mysteries in the success of neural networks is randomly initialized first order methods like gradient descent can achieve zero training loss even though the objective function is non-convex and non-smooth. This paper demystifies…