Generalizing Convolutional Neural Networks for Equivariance to Lie Groups on Arbitrary Continuous Data

Finzi, Marc; Stanton, Samuel C.; Izmailov, Pavel; Wilson, Andrew Gordon

doi:10.48550/arxiv.2002.12880

Cited by 29 publications

(52 citation statements)

References 21 publications

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“…Equivariant DL models have achieved remarkable success in learning image data (Cohen et al, 2019;Weiler & Cesa, 2019b;Cohen & Welling, 2016a;Chidester et al, 2018;Lenc & Vedaldi, 2015;Kondor & Trivedi, 2018;Bao & Song, 2019;Worrall et al, 2017;Cohen & Welling, 2016b;Finzi et al, 2020;Weiler et al, 2018b;Dieleman et al, 2016;Ghosh & Gupta, 2019;Sosnovik et al, 2020b).…”

Section: Equivariance and Invariancementioning

confidence: 99%

Approximately Equivariant Networks for Imperfectly Symmetric Dynamics

Wang¹,

Walters²,

Yu³

2022

Preprint

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Incorporating symmetry as an inductive bias into neural network architecture has led to improvements in generalization, data efficiency, and physical consistency in dynamics modeling. Methods such as CNNs or equivariant neural networks use weight tying to enforce symmetries such as shift invariance or rotational equivariance. However, despite the fact that physical laws obey many symmetries, real-world dynamical data rarely conforms to strict mathematical symmetry either due to noisy or incomplete data or to symmetry breaking features in the underlying dynamical system. We explore approximately equivariant networks which are biased towards preserving symmetry but are not strictly constrained to do so. By relaxing equivariance constraints, we find that our models can outperform both baselines with no symmetry bias and baselines with overly strict symmetry in both simulated turbulence domains and real-world multi-stream jet flow.

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Section: Equivariance and Invariancementioning

confidence: 99%

Approximately Equivariant Networks for Imperfectly Symmetric Dynamics

Wang¹,

Walters²,

Yu³

2022

Preprint

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“…The work of [10] also showed that by exploiting the Euler-Lagrange equation, it is possible to predict a controlled double pendulum -a system pertinent for controlled robots. A recent advance shows that Hamiltonian and Lagrangian NNs can be drastically improved if they are optimized over Cartesian coordinates with holonomic constraints [29].…”

Section: Related Workmentioning

confidence: 99%

Port-Hamiltonian Neural Networks for Learning Explicit Time-Dependent Dynamical Systems

Desai,

Mattheakis,

Sondak

et al. 2021

Preprint

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Accurately learning the temporal behavior of dynamical systems requires models with well-chosen learning biases. Recent innovations embed the Hamiltonian and Lagrangian formalisms into neural networks and demonstrate a significant improvement over other approaches in predicting trajectories of physical systems. These methods generally tackle autonomous systems that depend implicitly on time or systems for which a control signal is known apriori. Despite this success, many real world dynamical systems are non-autonomous, driven by time-dependent forces and experience energy dissipation. In this study, we address the challenge of learning from such non-autonomous systems by embedding the port-Hamiltonian formalism into neural networks, a versatile framework that can capture energy dissipation and time-dependent control forces. We show that the proposed port-Hamiltonian neural network can efficiently learn the dynamics of nonlinear physical systems of practical interest and accurately recover the underlying stationary Hamiltonian, time-dependent force, and dissipative coefficient. A promising outcome of our network is its ability to learn and predict chaotic systems such as the Duffing equation, for which the trajectories are typically hard to learn.

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“…For CNN we used 3 × 3 kernels and equivalently used n L = 9 for the number of L i in L-conv and random L-conv. We also used "LieConv" Finzi et al (2020) as a baseline (Fig. 10, brown).…”

Section: Experiments On Imagesmentioning

confidence: 99%

“…For the symmetry group in LieConv we used SE(3). We also used the default ResNet architecture provided by Finzi et al (2020) for both the one and two layer experiments. We turned off batch normalization, consistent with other experiments.…”

Section: Experiments On Imagesmentioning

confidence: 99%

Automatic Symmetry Discovery with Lie Algebra Convolutional Network

Dehmamy

Walters

Liu

et al. 2021

Preprint

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Existing equivariant neural networks for continuous groups require discretization or group representations. All these approaches require detailed knowledge of the group parametrization and cannot learn entirely new symmetries. We propose to work with the Lie algebra (infinitesimal generators) instead of the Lie group. Our model, the Lie algebra convolutional network (L-conv) can learn potential symmetries and does not require discretization of the group. We show that L-conv can serve as a building block to construct any group equivariant architecture. We discuss how CNNs and Graph Convolutional Networks are related to and can be expressed as L-conv with appropriate groups. We also derive the MSE loss for a single L-conv layer and find a deep relation with Lagrangians used in physics, with some of the physics aiding in defining generalization and symmetries in the loss landscape. Conversely, L-conv could be used to propose more general equivariant ansätze for scientific machine learning.Another approach taken in Finzi et al. ( 2020) approximates the integral over the Lie group by Monte Carlo sampling. However, their approach is quite involved, using the matrix logarithm and exponential to define a distance function used to obtain a sampling neighborhood for each point. Finzi et al.(2021) provides a unified approach for encoding continuous groups multi-layer perceptrons (MLP).Preprint. Under review.

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Generalizing Convolutional Neural Networks for Equivariance to Lie Groups on Arbitrary Continuous Data

Cited by 29 publications

References 21 publications

Approximately Equivariant Networks for Imperfectly Symmetric Dynamics

Approximately Equivariant Networks for Imperfectly Symmetric Dynamics

Port-Hamiltonian Neural Networks for Learning Explicit Time-Dependent Dynamical Systems

Automatic Symmetry Discovery with Lie Algebra Convolutional Network

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