Space-Time Continuous PDE Forecasting using Equivariant Neural Fields (2406.06660v1)

Abstract: Recently, Conditional Neural Fields (NeFs) have emerged as a powerful modelling paradigm for PDEs, by learning solutions as flows in the latent space of the Conditional NeF. Although benefiting from favourable properties of NeFs such as grid-agnosticity and space-time-continuous dynamics modelling, this approach limits the ability to impose known constraints of the PDE on the solutions -- e.g. symmetries or boundary conditions -- in favour of modelling flexibility. Instead, we propose a space-time continuous NeF-based solving framework that - by preserving geometric information in the latent space - respects known symmetries of the PDE. We show that modelling solutions as flows of pointclouds over the group of interest $G$ improves generalization and data-efficiency. We validated that our framework readily generalizes to unseen spatial and temporal locations, as well as geometric transformations of the initial conditions - where other NeF-based PDE forecasting methods fail - and improve over baselines in a number of challenging geometries.

• The paper introduces an innovative framework that integrates geometric symmetries into neural fields for continuous PDE forecasting.
• It employs equivariant neural ODEs to maintain latent space flow consistency under spatial transformations.
• Experiments on diverse geometries demonstrate improved MSE and exceptional generalization to unseen PDE configurations.

Overview of "Space-Time Continuous PDE Forecasting using Equivariant Neural Fields"

The paper "Space-Time Continuous PDE Forecasting using Equivariant Neural Fields" introduces an innovative framework designed for solving Partial Differential Equations (PDEs) that underscores the significance of incorporating equivariance constraints. Traditional methods for solving PDEs have relied on numerical solutions such as finite element or spectral methods; however, the paper seeks to leverage Deep Learning (DL) techniques, specifically Conditional Neural Fields (NeFs), to improve generalization and efficiency. A key tenet of this framework is the maintenance of known symmetries of the PDE within the latent spaces of the Conditional Neural Fields.

Geometric Equivariance in Neural Fields

The authors focus on integrating geometric properties, specifically symmetries, directly into the NeF architectures, resulting in Equivariant Neural Fields (ENFs). These fields are capable of accommodating the geometric transformations inherent in the PDE systems, such as rotations, translations, and periodic conditions. By employing weight-sharing schemes over bi-invariant attributes of input latitude-pose pairs, the framework respects symmetries characterized by groups like $\mathrm{SE(n)}$. For example, the paper illustrates that symmetries like full rotational invariance on the 2-sphere or periodic translations on the 2-torus can be minimally represented by appropriate bi-invariant attributes, which in turn optimize the learning process in PDE-related tasks.

Equivariant ODEs and Latent Space Dynamics

The introduction of neural ODEs that are equivariant plays a crucial role in the proposed methodology. When initial conditions are notably symmetrically transformed, equivariant ODEs allow latent space flows to develop into solutions that could transform reciprocally under the same symmetries. This is effectively achieved by parametrizing pose updates through the use of manifold-based logarithmic maps, which allows the system to project directly into a latent flow that maintains equivariances throughout temporal evolution.

Enhancing Latent Space Structure via Meta-Learning

Meta-learning constitutes a pivotal strategy for structuring the latent space of the ENFs. Traditional approaches, such as auto-decoding, involve numerous optimization steps for each individual input, which can lead to inefficiencies and difficult-to-navigate latent spaces. Instead, the authors utilize a method whereby latent states are initialized by a few meta-learned steps, consequently aligning them into a well-structured space. This ultimately simplifies the optimization path for equivariant neural ODEs, as demonstrated by the comparisons between latent visualizations using various training methodologies.

Experimental Verification and Results

The experimental setup spans multiple PDE systems across varying geometric configurations, such as the plane, 2-torus, 2-sphere, and 3D ball, showcasing the versatility and robustness of the proposed framework. This empirical analysis demonstrates noticeable improvements in MSE over both time horizons extrapolating beyond those seen during training and spatial generalization to unseen locations. A particular focus is given to ensuring that the PDE solutions respect underlying symmetries within the data, outperforming prior NeF-based approaches, such as DINo, and consistently achieving lower error rates even under challenging data conditions.

Implications and Future Directions

The implications of adopting such a framework are multifaceted, notably aiding in the further integration of machine learning techniques within traditional numerical methods for solving PDEs. Though this paper emphasizes symmetry preservation as a key feature, potential future work could explore extending the encapsulation of more complex empirical phenomena beyond current geometric considerations. Additionally, further enhancement of meta-learning strategies coupled with equivariant architectures can continue to reduce inference times and computational load, creating broader applicability outside conventional academic settings into real-world scenarios that demand rapid and precise PDE solutions. The approach also paves the way for a more homogeneous blend between data-driven learning paradigms and the longstanding analytical methods intrinsic to scientific computing.