Related papers: Filtered Partial Differential Equations: a robust …

We present a new scientific machine learning method that learns from data a computationally inexpensive surrogate model for predicting the evolution of a system governed by a time-dependent nonlinear partial differential equation (PDE), an…

We propose a methodology that combines generative latent diffusion models with physics-informed machine learning to generate solutions of parametric partial differential equations (PDEs) conditioned on partial observations, which includes,…

Physics informed neural networks (PINNs) have emerged as a powerful tool to provide robust and accurate approximations of solutions to partial differential equations (PDEs). However, PINNs face serious difficulties and challenges when…

Partial Differential Equations (PDE) are fundamental to model different phenomena in science and engineering mathematically. Solving them is a crucial step towards a precise knowledge of the behaviour of natural and engineered systems. In…

We present approximation theories and efficient training methods for derivative-informed Fourier neural operators (DIFNOs) with applications to PDE-constrained optimization. A DIFNO is an FNO trained by minimizing its prediction error…

Recent advances in deep learning have inspired numerous works on data-driven solutions to partial differential equation (PDE) problems. These neural PDE solvers can often be much faster than their numerical counterparts; however, each…

Neural networks are one tool for approximating non-linear differential equations used in scientific computing tasks such as surrogate modeling, real-time predictions, and optimal control. PDE foundation models utilize neural networks to…

Data-driven methods have recently made great progress in the discovery of partial differential equations (PDEs) from spatial-temporal data. However, several challenges remain to be solved, including sparse noisy data, incomplete candidate…

Solving partial differential equations (PDEs) is an important yet challenging task in fluid mechanics. In this study, we embed an improved Fourier series into neural networks and propose a physics-informed Fourier basis neural network…

Enhancing neural networks with knowledge of physical equations has become an efficient way of solving various physics problems, from fluid flow to electromagnetism. Graph neural networks show promise in accurately representing irregularly…

In this paper, we propose physics-informed neural operators (PINO) that combine training data and physics constraints to learn the solution operator of a given family of parametric Partial Differential Equations (PDE). PINO is the first…

Neural operators have shown great potential in surrogate modeling. However, training a well-performing neural operator typically requires a substantial amount of data, which can pose a major challenge in complex applications. In such…

Scientific machine learning has been successfully applied to inverse problems and PDE discovery in computational physics. One caveat concerning current methods is the need for large amounts of ("clean") data, in order to characterize the…

Partial differential equations (PDEs) play a crucial role in studying a vast number of problems in science and engineering. Numerically solving nonlinear and/or high-dimensional PDEs is often a challenging task. Inspired by the traditional…

Recent research has used deep learning to develop partial differential equation (PDE) models in science and engineering. The functional form of the PDE is determined by a neural network, and the neural network parameters are calibrated to…

Derivation of the probability density evolution provides invaluable insight into the behavior of many stochastic systems and their performance. However, for most real-time applica-tions, numerical determination of the probability density…

Surrogate modeling of costly mathematical models representing physical systems is challenging since it is typically not possible to create a large experimental design. Thus, it is beneficial to constrain the approximation to adhere to the…

We present a methodology combining neural networks with physical principle constraints in the form of partial differential equations (PDEs). The approach allows to train neural networks while respecting the PDEs as a strong constraint in…

Neural operators have emerged as promising surrogate models for solving partial differential equations (PDEs), but struggle to generalise beyond training distributions and are often constrained to a fixed temporal discretisation. This work…

This work presents a physics-informed deep learning-based super-resolution framework to enhance the spatio-temporal resolution of the solution of time-dependent partial differential equations (PDE). Prior works on deep learning-based…