Learning Conserved Quantities and Symmetries in Hamiltonian Neural Networks using Noether's Theorem and Bayesian Model Selection
Core Concepts
This paper introduces Noether's Razor, a novel method for automatically discovering symmetries in dynamical systems by leveraging Noether's theorem to parameterize symmetries as learnable conserved quantities within Hamiltonian Neural Networks and employing Bayesian model selection to learn these symmetries directly from training data.
Abstract

Bibliographic Information: van der Ouderaa, T. F. A., van der Wilk, M., de Haan, P. (2024). Noether’s razor: Learning Conserved Quantities. Advances in Neural Information Processing Systems, 37.

Research Objective: This paper aims to address the challenge of automatically discovering symmetries in dynamical systems, which can significantly improve the generalization and performance of machine learning models.

Methodology: The authors propose Noether's Razor, a method that combines Noether's theorem with Bayesian model selection. They parameterize symmetries in Hamiltonian Neural Networks (HNNs) using learnable conserved quantities. By incorporating these conserved quantities into the prior of the HNN and optimizing an approximate marginal likelihood, the method learns the symmetries directly from training data. The authors derive a variational lower bound on the marginal likelihood to make the optimization tractable.

Key Findings: The authors demonstrate the effectiveness of Noether's Razor on various dynamical systems, including nsimple harmonic oscillators and nbody systems. Their results show that the method successfully identifies the correct conserved quantities and corresponding symmetry groups (U(n) for oscillators and SE(n) for nbody systems).

Main Conclusions: Noether's Razor offers a powerful and principled approach to automatically discover symmetries in dynamical systems from data. The method matches the performance of models with oracle symmetries and outperforms vanilla HNNs, leading to improved generalization and predictive accuracy on test data.

Significance: This research significantly contributes to the field of physicsinformed machine learning and symmetry discovery. By integrating fundamental physics principles with machine learning, it paves the way for developing more robust and generalizable models for complex dynamical systems.

Limitations and Future Research: The current work focuses on quadratic conserved quantities. Exploring richer function classes for conserved quantities, such as neural networks, could be a promising direction for future research. However, this would require addressing potential overfitting issues through appropriate priors or regularization techniques.
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Noether's razor: Learning Conserved Quantities
Stats
The learned symmetry for the simple harmonic oscillator model has a better ELBO on the training set compared to the vanilla HNN model.
The symmetry learning method for the simple harmonic oscillator achieves a test MSE of 0.002, matching the performance of the model with the correct SO(2) symmetry builtin.
For the nharmonic oscillators, the matrix of learned symmetries consistently exhibits n² nonzero singular values, indicating the correct learning of the U(n) symmetry.
The first n² singular vectors for the nharmonic oscillators lie in the ground truth subspace of generators with a measured parallelness vᵢ > 0.99.
In the 3body system experiment, the method correctly learns 7 singular values with λᵢ > 0.05, and the associated singular vectors lie in the ground truth subspace with vᵢ > 0.95.
Quotes
"Symmetries provide strong inductive biases, effectively reducing the volume of the hypothesis space."
"Rather than strictly constraining a model to certain symmetries, recent works have explored whether invariance and equivariance symmetries in machine learning models can also be automatically learned from data."
"We propose to use Noether’s theorem [Noether, 1918] to parameterise symmetries in Hamiltonian machine learning models in terms of their conserved quantities."
"This work, however, provides evidence that it is also possible to perform approximate Bayesian model selection using VI in deep neural networks, which we deem an interesting observation in its own right."
Deeper Inquiries
How might Noether's Razor be extended to handle systems with partially observed or noisy data, where the true symmetries might be obscured?
Handling partially observed or noisy data presents a significant challenge for Noether's Razor, as the core assumption of clean, fully observed trajectories in phase space may no longer hold. Here are potential strategies to address this:
Robust Symmetrization: Instead of directly averaging over the transformed data points using the learned symmetries (as in Equation 5), we could explore more robust averaging techniques. These could include:
Medianbased averaging: Less sensitive to outliers compared to the mean.
Weighted averaging: Assign weights to data points based on their estimated reliability or confidence. This could be derived from the noise model of the data or through techniques like Gaussian Process latent variable models.
Latent Variable Models: For partially observed systems, we can incorporate latent variables to represent the missing information.
Variational Autoencoders (VAEs): VAEs can learn a lowerdimensional latent representation of the system that captures the underlying dynamics. Noether's Razor could be applied in this latent space, potentially making the symmetries more apparent.
StateSpace Models: These models are naturally suited for systems with noisy or partially observed data. We could integrate Noether's Razor into the statespace framework, perhaps by incorporating the learned symmetries into the state transition function.
Denoising Techniques: Preprocess the data using denoising techniques specifically designed for time series data. This could involve:
Wavelet denoising: Effective at separating noise from signal in time series.
** Kalman filtering:** Suitable for systems with known dynamics, it can estimate the true state of the system from noisy measurements.
Weakly Supervised Learning: If we have some prior knowledge about the types of symmetries that might be present, we can incorporate this information as weak supervision. For example, if we suspect a rotational symmetry, we could encourage the model to learn conserved quantities that resemble angular momentum.
It's important to note that these extensions would likely increase the complexity of the model and the optimization process. Further research is needed to determine the most effective strategies for applying Noether's Razor to realworld systems with noisy and incomplete data.
Could the reliance on quadratic conserved quantities limit the applicability of Noether's Razor to systems with more complex, nonlinear symmetries?
Yes, the current formulation of Noether's Razor, by design, focuses on quadratic conserved quantities. This choice offers computational advantages, as the symmetry transformations and their flows have closedform solutions (Equation 4). However, this reliance on quadratic forms indeed limits its applicability to systems possessing more complex, nonlinear symmetries.
Here's why this is a limitation and potential ways to address it:
Nonlinear symmetries are abundant: Many physical systems exhibit symmetries that cannot be expressed through quadratic conserved quantities. Examples include:
The Kepler problem: While possessing conserved quantities like angular momentum (quadratic), the shape of the orbit (elliptical, parabolic, hyperbolic) is dictated by the LaplaceRungeLenz vector, which has a nonlinear relationship with position and momentum.
Field theories: Symmetries like gauge invariance in electromagnetism or the diffeomorphism invariance in general relativity involve transformations that are inherently nonlinear.
Overcoming the Limitation:
HigherOrder Conserved Quantities: Extend the framework to include higherorder polynomials in the conserved quantities. This would allow for the representation of more complex symmetries, but at the cost of increased computational complexity.
Neural Network Representation: Instead of explicitly defining the form of the conserved quantities, we could use neural networks to learn them directly from data. This offers flexibility but requires careful regularization to prevent overfitting and ensure the learned quantities genuinely represent symmetries.
Hybrid Approaches: Combine the current approach with additional constraints or inductive biases tailored to the specific nonlinear symmetries expected in the system. This could involve incorporating domain knowledge or using specialized neural network architectures.
Exploring these extensions is crucial for broadening the applicability of Noether's Razor to a wider range of dynamical systems.
What are the potential implications of automatically discovering symmetries in dynamical systems for other scientific domains beyond physics, such as biology or chemistry?
The ability to automatically discover symmetries in dynamical systems using techniques like Noether's Razor holds exciting implications for various scientific domains beyond physics, including biology and chemistry. Here are some potential applications:
Biology:
Systems Biology: Biological systems, from gene regulatory networks to ecosystems, are often characterized by complex interactions and emergent behaviors. Identifying symmetries in these systems could:
Simplify models: Reduce the complexity of models by identifying conserved quantities and invariant properties.
Uncover design principles: Reveal fundamental organizational principles governing biological systems.
Predict system behavior: Improve the accuracy and robustness of predictions about system dynamics.
Drug Discovery: Symmetries in molecular interactions and biological pathways could be exploited for drug design. For example:
Identify drug targets: Focus on molecules or pathways exhibiting specific symmetries that are crucial for disease progression.
Design more effective drugs: Develop drugs that exploit or preserve essential symmetries in biological systems.
Evolutionary Biology: Understanding symmetries in biological forms and developmental processes could provide insights into:
Constraints on evolution: Identify the role of symmetries in shaping evolutionary trajectories.
Origins of biological complexity: Explore how symmetries contribute to the emergence of complex structures and functions.
Chemistry:
Reaction Kinetics: Discovering symmetries in chemical reaction networks could:
Simplify reaction mechanisms: Identify conserved quantities and reduce the number of independent variables needed to describe a reaction.
Predict reaction pathways: Determine the most likely reaction pathways based on symmetry considerations.
Design new catalysts: Develop catalysts that exploit symmetries to enhance reaction rates or selectivity.
Materials Science: Symmetries play a crucial role in determining the properties of materials. Automatic symmetry discovery could aid in:
Designing new materials: Create materials with desired properties by tailoring their symmetries.
Predicting material behavior: Understand and predict the behavior of materials under different conditions based on their symmetries.
General Implications:
Datadriven discovery: Noether's Razor and similar techniques enable datadriven discovery of fundamental principles in complex systems, potentially leading to new insights and breakthroughs.
Interdisciplinary research: The application of symmetry principles across disciplines could foster collaboration and accelerate scientific progress.
Improved understanding of complex systems: By uncovering hidden symmetries, we can gain a deeper understanding of the underlying order and organization of complex systems in various scientific domains.