Related papers: Numerical Solution of Mixed-Dimensional PDEs Using…
Partial differential equations (PDEs) play a foundational role in modeling physical phenomena. This study addresses the challenging task of determining variable coefficients within PDEs from measurement data. We introduce a novel neural…
Neural networks have shown promising potential in accelerating the numerical simulation of systems governed by partial differential equations (PDEs). Different from many existing neural network surrogates operating on high-dimensional…
Partial differential equations (PDEs) have become an essential tool for modeling complex physical systems. Such equations are typically solved numerically via mesh-based methods, such as finite element methods, with solutions over the…
Coupled 3D-1D problems arise in many practical applications, in an attempt to reduce the computational burden in simulations where cylindrical inclusions with a small section are embedded in a much larger domain. Nonetheless the resolution…
We consider a mixed dimensional elliptic partial differential equation posed in a bulk domain with a large number of embedded interfaces. In particular, we study well-posedness of the problem and regularity of the solution. We also propose…
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…
The recent rise of deep learning has led to numerous applications, including solving partial differential equations using Physics-Informed Neural Networks. This approach has proven highly effective in several academic cases. However, their…
This article explores operator learning models that can deduce solutions to partial differential equations (PDEs) on arbitrary domains without requiring retraining. We introduce two innovative models rooted in boundary integral equations…
The evolution of dynamical systems is generically governed by nonlinear partial differential equations (PDEs), whose solution, in a simulation framework, requires vast amounts of computational resources. In this work, we present a novel…
Parabolic partial differential equations (PDEs) and backward stochastic differential equations (BSDEs) are key ingredients in a number of models in physics and financial engineering. In particular, parabolic PDEs and BSDEs are fundamental…
The multilevel Schwarz preconditioner is one of the most popular parallel preconditioners for enhancing convergence and improving parallel efficiency. However, its parallel implementation on arbitrary unstructured triangular/tetrahedral…
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…
We consider parameterized variational inverse problems that are constrained by partial differential equations (PDEs). We seek to efficiently compute the solution of the inverse problem when auxiliary model parameters, which appear in the…
In the present study, we consider the Extra-Membrane-Intra model (EMI) for the simulation of excitable tissues at the cellular level. We provide the (possibly large) system of partial differential equations (PDEs), equipped with ad hoc…
We revisit gradient-based optimization for infinite projected entangled pair states (iPEPS), a tensor network ansatz for simulating many-body quantum systems. This approach is hindered by two major challenges: the high computational cost of…
In this paper, we propose a novel algorithm called Neuron-wise Parallel Subspace Correction Method (NPSC) for the finite neuron method that approximates numerical solutions of partial differential equations (PDEs) using neural network…
Many physical processes such as weather phenomena or fluid mechanics are governed by partial differential equations (PDEs). Modelling such dynamical systems using Neural Networks is an active research field. However, current methods are…
Stochastic partial differential equations (SPDEs) are the mathematical tool of choice for modelling spatiotemporal PDE-dynamics under the influence of randomness. Based on the notion of mild solution of an SPDE, we introduce a novel neural…
Neural operators have emerged as an efficient paradigm for solving PDEs, overcoming the limitations of traditional numerical methods and significantly improving computational efficiency. However, due to the diversity and complexity of PDE…
Metasurfaces, typically realized as arrays of nanopillars, transform electromagnetic (EM) fields depending on their geometry and spatial arrangement. For solving the inverse problem of designing new metasurfaces that transform EM fields in…