Latent Neural PDE Solver: a reduced-order modelling framework for partial differential equations

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 discretized fields, we propose to learn the dynamics of the system in the latent space with much coarser discretizations. In our proposed framework - Latent Neural PDE Solver (LNS), a non-linear autoencoder is first trained to project the full-order representation of the system onto the mesh-reduced space, then a temporal model is trained to predict the future state in this mesh-reduced space. This reduction process simplifies the training of the temporal model by greatly reducing the computational cost accompanying a fine discretization. We study the capability of the proposed framework and several other popular neural PDE solvers on various types of systems including single-phase and multi-phase flows along with varying system parameters. We showcase that it has competitive accuracy and efficiency compared to the neural PDE solver that operates on full-order space.

HNO: Hyena Neural Operator for solving PDEs

Numerically solving partial differential equations (PDEs) typically requires fine discretization to resolve necessary spatiotemporal scales, which can be computationally expensive. Recent advances in deep learning have provided a new approach to solving PDEs that involves the use of neural operators. Neural operators are neural network architectures that learn mappings between function spaces and have the capability to solve partial differential equations based on data. This study utilizes a novel neural operator called Hyena, which employs a long convolutional filter that is parameterized by a multilayer perceptron. The Hyena operator is an operation that enjoys sub-quadratic complexity and state space model to parameterize long convolution that enjoys global receptive field. This mechanism enhances the model's comprehension of the input's context and enables data-dependent weight for different PDE instances. To measure how effective the layers are in solving PDEs, we conduct experiments on Burger's equation and Navier Stokes equation. Our findings indicate Hyena Neural operator can serve as an efficient and accurate model for learning PDEs' solution operator. The data and code used can be found at: https://github.com/Saupatil07/Hyena-Neural-Operator