Related papers: From Fourier to Neural ODEs: Flow Matching for Mod…

Accurate short-term predictions of phase-resolved water wave conditions are crucial for decision-making in ocean engineering. However, the initialization of remote-sensing-based wave prediction models first requires a reconstruction of wave…

Many engineered physical processes exhibit nonlinear but asymptotically stable dynamics that converge to a finite set of equilibria determined by control inputs. Identifying such systems from data is challenging: stable dynamics provide…

Learning probabilistic surrogates for partial differential equations remains challenging in data-scarce regimes: neural operators require large amounts of high-fidelity data, while generative approaches typically sacrifice resolution…

Deep learning has become a pivotal technology in fields such as computer vision, scientific computing, and dynamical systems, significantly advancing these disciplines. However, neural Networks persistently face challenges related to…

Embedding physical knowledge into neural network (NN) training has been a hot topic. However, when facing the complex real-world, most of the existing methods still strongly rely on the quantity and quality of observation data. Furthermore,…

Neural ordinary differential equations (neural ODEs) can effectively learn dynamical systems from time series data, but their behavior on graph-structured data remains poorly understood, especially when applied to graphs with different size…

We propose an extended Fourier neural operator (FNO) architecture for learning state and linear quadratic additive optimal control of systems governed by partial differential equations. Using the Ehrenpreis-Palamodov fundamental principle,…

Owing to the remarkable development of deep learning technology, there have been a series of efforts to build deep learning-based climate models. Whereas most of them utilize recurrent neural networks and/or graph neural networks, we design…

Inspired by the ubiquitous use of differential equations to model continuous dynamics across diverse scientific and engineering domains, we propose a novel and intuitive approach to continuous sequence modeling. Our method interprets…

Finite element analysis (FEA) has been widely used to generate simulations of complex and nonlinear systems. Despite its strength and accuracy, the limitations of FEA can be summarized into two aspects: a) running high-fidelity FEA often…

Artificial neural networks, widely recognised for their role in machine learning, are now transforming the study of ordinary differential equations (ODEs), bridging data-driven modelling with classical dynamical systems and enabling the…

Neural PDE solvers are increasingly used as learned surrogates for families of partial differential equations, where the key machine learning challenge is not only interpolation on a fixed benchmark distribution but generalization under…

Modeling future traffic conditions often relies heavily on complex spatial-temporal neural networks to capture spatial and temporal correlations, which can overlook the inherent noise in the data. This noise, often manifesting as unexpected…

We introduce a method that combines neural operators, physics-informed machine learning, and standard numerical methods for solving PDEs. The proposed approach extends each of the aforementioned methods and unifies them within a single…

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…

The numerical solution of partial differential equations (PDEs) is fundamental to scientific and engineering computing. In the presence of strong anisotropy, material heterogeneity, and complex geometries, however, classical iterative…

We introduce the Fourier Learning Machine (FLM), a neural network (NN) architecture designed to represent a multidimensional nonharmonic Fourier series. The FLM uses a simple feedforward structure with cosine activation functions to learn…

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,…

Deep Learning has emerged as one of the most significant innovations in machine learning. However, a notable limitation of this field lies in the ``black box" decision-making processes, which have led to skepticism within groups like…

Deep neural network models have shown a great potential in accelerating the simulation of fluid dynamic systems. Once trained, these models can make inference within seconds, thus can be extremely efficient. However, they suffer from a…