Related papers: Component Fourier Neural Operator for Singularly P…

Neural operators such as the Fourier Neural Operator (FNO) have been shown to provide resolution-independent deep learning models that can learn mappings between function spaces. For example, an initial condition can be mapped to the…

Neural operators provide fast surrogate models for time-dependent partial differential equations, but their standard autoregressive use usually assumes that the instantaneous field $u(t,\cdot)$ is a complete state. This assumption fails for…

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…

We study learning weak solutions to nonlinear hyperbolic partial differential equations (H-PDE), which have been difficult to learn due to discontinuities in their solutions. We use a physics-informed variant of the Fourier Neural Operator…

We present a Fourier neural network (FNN) that can be mapped directly to the Fourier decomposition. The choice of activation and loss function yields results that replicate a Fourier series expansion closely while preserving a…

Interfacial dynamics underlie a wide range of phenomena, including phase transitions, microstructure coarsening, pattern formation, and thin-film growth, and are typically described by stiff, time-dependent nonlinear partial differential…

Spectral methods for solving partial differential equations (PDEs) and stochastic partial differential equations (SPDEs) often use Fourier or polynomial spectral expansions on either uniform and non-uniform grids. However, while very widely…

A novel approach to approximate solutions of Stochastic Differential Equations (SDEs) by Deep Neural Networks is derived and analysed. The architecture is inspired by the notion of Deep Operator Networks (DeepONets), which is based on…

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…

Classical sequential models employed in time-series prediction rely on learning the mappings from the past to the future instances by way of a hidden state. The Hidden states characterise the historical information and encode the required…

Partial differential equations (PDEs) underlie our understanding and prediction of natural phenomena across numerous fields, including physics, engineering, and finance. However, solving parametric PDEs is a complex task that necessitates…

We present a novel framework combining Deep Operator Networks (DeepONets) with Physics-Informed Neural Networks (PINNs) to solve partial differential equations (PDEs) and estimate their unknown parameters. By integrating data-driven…

This study proposes a self-optimization physics-informed Fourier-features randomized neural network (SO-PIFRNN) framework, which significantly improves the numerical solving accuracy of PDEs through hyperparameter optimization mechanism.…

Scientific machine learning has seen significant progress with the emergence of operator learning. However, existing methods encounter difficulties when applied to problems on unstructured grids and irregular domains. Spatial graph neural…

We propose a novel neural preconditioned Newton (NP-Newton) method for solving parametric nonlinear systems of equations. To overcome the stagnation or instability of Newton iterations caused by unbalanced nonlinearities, we introduce a…

Deep learning-based surrogate models have been widely applied in geological carbon storage (GCS) problems to accelerate the prediction of reservoir pressure and CO2 plume migration. Large amounts of data from physics-based numerical…

Neural operators have emerged as a powerful data-driven paradigm for solving partial differential equations (PDEs), while their accuracy and scalability are still limited, particularly on irregular domains where fluid flows exhibit rich…

Solving parametric Partial Differential Equations (PDEs) for a broad range of parameters is a critical challenge in scientific computing. To this end, neural operators, which \textcolor{black}{predicts the PDE solution with variable PDE…

With the recent rise of neural operators, scientific machine learning offers new solutions to quantify uncertainties associated with high-fidelity numerical simulations. Traditional neural networks, such as Convolutional Neural Networks…

The computational efficiency of many neural operators, widely used for learning solutions of PDEs, relies on the fast Fourier transform (FFT) for performing spectral computations. As the FFT is limited to equispaced (rectangular) grids,…