Neural Canonical Transformation with Symplectic Flows
This work addresses the challenge of solving complex Hamiltonian systems in physics and machine learning, offering a novel method for canonical transformations with potential applications in molecular dynamics and data compression, though it appears incremental as it builds on existing symplectic and neural network techniques.
The authors tackled the problem of simplifying classical Hamiltonian systems by constructing flexible canonical transformations using symplectic neural networks, transforming physical variables into a latent representation with an independent harmonic oscillator Hamiltonian and factorized Gaussian distribution, and demonstrated its application on toy problems and real-world tasks like identifying slow collective modes in alanine dipeptide and compressing the MNIST dataset.
Canonical transformation plays a fundamental role in simplifying and solving classical Hamiltonian systems. We construct flexible and powerful canonical transformations as generative models using symplectic neural networks. The model transforms physical variables towards a latent representation with an independent harmonic oscillator Hamiltonian. Correspondingly, the phase space density of the physical system flows towards a factorized Gaussian distribution in the latent space. Since the canonical transformation preserves the Hamiltonian evolution, the model captures nonlinear collective modes in the learned latent representation. We present an efficient implementation of symplectic neural coordinate transformations and two ways to train the model. The variational free energy calculation is based on the analytical form of physical Hamiltonian. While the phase space density estimation only requires samples in the coordinate space for separable Hamiltonians. We demonstrate appealing features of neural canonical transformation using toy problems including two-dimensional ring potential and harmonic chain. Finally, we apply the approach to real-world problems such as identifying slow collective modes in alanine dipeptide and conceptual compression of the MNIST dataset.