Incorporating Symmetry into Deep Dynamics Models for Improved Generalization (2002.03061v4)

Abstract: Recent work has shown deep learning can accelerate the prediction of physical dynamics relative to numerical solvers. However, limited physical accuracy and an inability to generalize under distributional shift limit its applicability to the real world. We propose to improve accuracy and generalization by incorporating symmetries into convolutional neural networks. Specifically, we employ a variety of methods each tailored to enforce a different symmetry. Our models are both theoretically and experimentally robust to distributional shift by symmetry group transformations and enjoy favorable sample complexity. We demonstrate the advantage of our approach on a variety of physical dynamics including Rayleigh B\'enard convection and real-world ocean currents and temperatures. Compared with image or text applications, our work is a significant step towards applying equivariant neural networks to high-dimensional systems with complex dynamics. We open-source our simulation, data, and code at \url{https://github.com/Rose-STL-Lab/Equivariant-Net}.

• The paper integrates symmetry principles via Noether's Law into CNNs to improve generalization in deep dynamics modeling.
• It achieves significant performance gains with an average 31% and maximum 78% reduction in energy error over conventional methods.
• The work lays the foundation for future research on equivariant networks in fields like fluid dynamics and material science.

Incorporating Symmetry in Deep Dynamics Models for Enhanced Generalization

The paper "Incorporating Symmetry into Deep Dynamics Models for Improved Generalization" by Rui Wang, Robin Walters, and Rose Yu offers a significant contribution to the application of deep learning in modeling physical dynamics systems through the novel integration of symmetry principles. This work is particularly focused on addressing challenges related to limited physical accuracy and poor generalization under distributional shifts inherent in conventional deep learning models used for dynamical systems predictions.

Overview and Methodological Approach

Deep learning models have shown potential to accelerate physical simulations but have often been hampered by issues related to generalization and physical accuracy. These limitations are rooted, to some extent, in the absence of canonical frames of reference in physical data, which creates difficulties in aligning out-of-distribution test data with training data. Additionally, inaccuracies in modeling may yield outputs that deviate from the true spatial energy distributions.

The authors propose a solution by incorporating symmetries into convolutional neural networks. They leverage Noether's Law, which correlates conserved quantities with symmetry groups, to enhance physical accuracy and generalization capabilities. The paper defines equivariance, where a function's transformation by a symmetry group results in an equivalent transformation of its output, and leverages this definition to construct neural networks that respect the symmetrical properties of the physical systems they aim to model.

The symmetries considered include translation, rotation, uniform motion (Galilean invariance), and scale invariance. Each type of symmetry is addressed with tailored methodologies, ranging from the implementation of $E(2)-CNN$ frameworks for rotation invariance to group correlation methods for resolving scale invariance.

Experimental Evaluation and Results

This approach was empirically tested against simulated datasets of Rayleigh-Bénard convection and real-world ocean current and temperature data. The models demonstrated enhanced generalization abilities and physical consistency compared to baselines. When symmetry groups such as translation, rotation, uniform motion, and scaling were incorporated, models manifested reduced energy spectrum error, showing improvements with an average reduction of $31\%$ and a maximum reduction of $78\%$ in energy error over conventional methods under no distributional shift.

Implications and Future Work

The implications of integrating symmetry into deep dynamics models are twofold: On a practical level, this enhances the ability of models to generalize to unseen data, a common challenge in real-world applications where data may not be uniformly distributed or aligned. Theoretically, this demonstrates the utility of incorporating physical laws into neural network architectures, opening paths for further research in equivariant networks not only in fluid dynamics but also potentially extending to complex systems within fields like molecular dynamics and material science.

Future work could refine these techniques, overcoming current limitations such as the computational demands of scale-equivariant models, and exploring unified models that incorporate the full symmetry group of the Navier-Stokes equations. Thorough exploration in this area may contribute to the development of more robust models that simulate physical systems with greater fidelity and breadth of applicability.

This paper represents a meaningful stride in bridging physics-informed principles with deep learning practices, aiming for models that not only predict but respect the underlying physical symmetry and conserve properties within their predictions.