Related papers: Autoregression-Free Neural Operators for Time-Depe…

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…

Neural Operators (NOs) are a leading method for surrogate modeling of partial differential equations. Unlike traditional neural networks, which approximate individual functions, NOs learn the mappings between function spaces. While NOs have…

Neural Ordinary Differential Equations (ODEs) are elegant reinterpretations of deep networks where continuous time can replace the discrete notion of depth, ODE solvers perform forward propagation, and the adjoint method enables efficient,…

We propose the Factorized Fourier Neural Operator (F-FNO), a learning-based approach for simulating partial differential equations (PDEs). Starting from a recently proposed Fourier representation of flow fields, the F-FNO bridges the…

Unsupervised learning with functional data is an emerging paradigm of machine learning research with applications to computer vision, climate modeling and physical systems. A natural way of modeling functional data is by learning operators…

Neural operators generalize neural networks to learn mappings between function spaces from data. They are commonly used to learn solution operators of parametric partial differential equations (PDEs) or propagators of time-dependent PDEs.…

Uncertainty and delayed reactions in human driving behavior lead to stop-and-go traffic congestion on freeways. The freeway traffic dynamics are governed by the Aw-Rascle-Zhang (ARZ) traffic Partial Differential Equation (PDE) models with…

Neural operators have emerged as a powerful, data-driven paradigm for learning solution operators of partial differential equations (PDEs). State-of-the-art architectures, such as the Fourier Neural Operator (FNO), have achieved remarkable…

Autoregressive next-step prediction models have become the de-facto standard for building data-driven neural solvers to forecast time-dependent partial differential equations (PDEs). Denoise training that is closely related to diffusion…

Although very successfully used in conventional machine learning, convolution based neural network architectures -- believed to be inconsistent in function space -- have been largely ignored in the context of learning solution operators of…

Accurately learning solution operators for time-dependent partial differential equations (PDEs) from sparse and irregular data remains a challenging task. Recurrent DeepONet extensions inherit the discrete-time limitations of…

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…

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…

Current time-series forecasting models are primarily based on transformer-style neural networks. These models achieve long-term forecasting mainly by scaling up the model size rather than through genuinely autoregressive (AR) rollout. From…

Neural operators for time-dependent PDEs face a structural tension: spectral architectures (FNO and descendants) inherit exponential rollout-error growth from their one-step Lipschitz constant, while hierarchical U-Net operators trade…

Neural operators (NO) are discretization invariant deep learning methods with functional output and can approximate any continuous operator. NO have demonstrated the superiority of solving partial differential equations (PDEs) over other…

Predicting the evolution of complex systems governed by partial differential equations (PDEs) remains challenging, especially for nonlinear, chaotic behaviors. This study introduces Koopman-inspired Fourier Neural Operators (kFNO) and…

By learning the mappings between infinite function spaces using carefully designed neural networks, the operator learning methodology has exhibited significantly more efficiency than traditional methods in solving complex problems such as…

Traditional approaches to stabilizing hyperbolic PDEs, such as PDE backstepping, often encounter challenges when dealing with high-dimensional or complex nonlinear problems. Their solutions require high computational and analytical costs.…

A wide range of scientific problems, such as those described by continuous-time dynamical systems and partial differential equations (PDEs), are naturally formulated on function spaces. While function spaces are typically…