Clifford Neural Layers for PDE Modeling (2209.04934v2)

Abstract: Partial differential equations (PDEs) see widespread use in sciences and engineering to describe simulation of physical processes as scalar and vector fields interacting and coevolving over time. Due to the computationally expensive nature of their standard solution methods, neural PDE surrogates have become an active research topic to accelerate these simulations. However, current methods do not explicitly take into account the relationship between different fields and their internal components, which are often correlated. Viewing the time evolution of such correlated fields through the lens of multivector fields allows us to overcome these limitations. Multivector fields consist of scalar, vector, as well as higher-order components, such as bivectors and trivectors. Their algebraic properties, such as multiplication, addition and other arithmetic operations can be described by Clifford algebras. To our knowledge, this paper presents the first usage of such multivector representations together with Clifford convolutions and Clifford Fourier transforms in the context of deep learning. The resulting Clifford neural layers are universally applicable and will find direct use in the areas of fluid dynamics, weather forecasting, and the modeling of physical systems in general. We empirically evaluate the benefit of Clifford neural layers by replacing convolution and Fourier operations in common neural PDE surrogates by their Clifford counterparts on 2D Navier-Stokes and weather modeling tasks, as well as 3D Maxwell equations. For similar parameter count, Clifford neural layers consistently improve generalization capabilities of the tested neural PDE surrogates. Source code for our PyTorch implementation is available at https://microsoft.github.io/cliffordlayers/.

• The paper introduces Clifford algebras into neural architectures, capturing scalar, vector, and bivector interactions for enhanced PDE modeling.
• The paper integrates Clifford convolutions and Fourier transforms into models like CResNet and CFNO, achieving lower rollout losses and increased stability.
• The paper demonstrates improved generalization in complex tasks such as Navier-Stokes, weather prediction, and Maxwell equations, especially with limited training data.

Clifford Neural Layers for PDE Modeling: An Overview

The research paper "Clifford Neural Layers for PDE Modeling" proposes an innovative approach to improving the modeling capabilities of neural networks in solving Partial Differential Equations (PDEs). The research team utilizes Clifford algebras to encapsulate scalar, vector, and higher-order components into multivectors, which are then processed using Clifford neural layers. The integration of Clifford convolutions and Clifford Fourier transforms into neural networks marks the paper's primary contribution, demonstrating enhanced generalization capabilities on relevant tasks such as the Navier-Stokes equations, weather prediction, and Maxwell's equations.

The exploration into using Clifford algebras arises from a fundamental limitation observed in existing neural PDE surrogates: the inability to exploit potential correlations between various field components effectively. Conventional methods indiscriminately stack scalar and vector components along a channel dimension without considering their geometric and arithmetic relations, leading to a loss of essential inductive bias. Multivectors, as represented by Clifford algebras, provide a more holistic view by accounting for scalar, vector, and bivector interactions, and allow for operations such as convolution to be expressed in these terms.

Empirical evaluations were conducted across tasks that included the simulation of 2D Navier-Stokes dynamics, weather modeling using shallow water equations, and 3D Maxwell equations. Each task benefits from the nuanced handling of field components offered by Clifford neural layers. In comparison to baseline architectures, Clifford-based architectures demonstrated consistent improvements in generalization, especially in datasets with limited training data.

For ResNet-like architectures, models enhanced with Clifford layers, coined CResNet, along with a variant focusing on rotational equitability, termed CResNet_rot, yielded superior results compared to the baseline ResNet models. Specifically, they exhibited lower rollout losses, suggesting a more stable prediction across time steps. On the Fourier Neural Operator (FNO) side, introducing Clifford Fourier layers (CFNOs) into the architecture also showed marked improvements over traditional FNOs by optimizing frequency domain interactions via multivector components.

This paper's implications extend to fields relying heavily on complex PDEs, such as fluid dynamics and electromagnetics. The inherent ability of Clifford layers to preserve and utilize geometric relationships paves the way for more efficient and accurate simulations. The potential for future development is vast; extensions into implicit neural representations, improvements in computational efficiency, and further exploration into combining Clifford models with physics-informed networks could underscore the trajectory of the research domain.

In conclusion, the paper establishes the groundwork for an impactful integration of Clifford algebras into neural network design, pushing the envelope on how deep learning can be applied to scientific modeling. The research opens up avenues for leveraging these algebraic structures to solve a broader set of scientific challenges, advancing the state-of-the-art in simulation accuracy and computational feasibility.