Variational autoencoders with latent high-dimensional steady geometric flows for dynamics

2020-04-16 · Source: stat.ML updates on arXiv.org · Field: Technology & Digital — Artificial Intelligence & Machine Learning, Mathematics & Computational Sciences, Robotics & Autonomous Systems · Depth: Expert, extended

Summary

A new framework, Variational Autoencoders with Dynamical Latent Manifolds (VAE-DLM), is introduced, applying Riemannian approaches to VAEs for PDE-type ambient data. VAE-DLM learns manifold geometries in the latent space, subject to a linear geometric flow with a steady-state regularizing term. This method reformulates the traditional evidence lower bound (ELBO) loss and requires only automatic differentiation of one time derivative, making it computationally efficient for moderately high dimensions (up to intrinsic dimension 15). The approach demonstrates robust learning, outperforming standard VAEs by up to 25% reduction in out-of-distribution (OOD) error, particularly for PDEs with minimal variation in late times, such as Burger's, Allen-Cahn, and modified porous medium equations. The architecture utilizes a modified multi-layer perceptron with tanh activations for the encoder-decoder.

Key takeaway

For Machine Learning Engineers or Research Scientists developing models for physical systems, you should consider integrating geometric flows into your VAE latent spaces. This VAE-DLM framework offers a computationally efficient method to achieve significant out-of-distribution error reduction, up to 25%, especially when dealing with steady-state partial differential equations. Explore its application to enhance the robustness and generalization capabilities of your dynamic models.

Key insights

VAE-DLM integrates geometric flows into latent spaces for robust learning of PDE dynamics, reducing OOD error.

Principles

• Geometric latent dynamics improve VAE robustness and generalization.
• Linear geometric flows with steady-state terms prevent manifold degeneracy.
• Physics-informed approaches enable efficient high-dimensional solutions.

Method

VAE-DLM redevelops the ELBO with a prior in parameterization space, learning manifold geometries subject to a linear geometric flow with a steady-state regularizer, solvable via automatic differentiation of one time derivative.

In practice

• Apply VAE-DLM to steady-state PDE dynamics for improved OOD robustness.
• Use intrinsic latent dimensions of 6-10 for effective representation.
• Employ modified MLP with tanh activations for encoder and decoder networks.

Topics

• Variational Autoencoders
• Geometric Deep Learning
• Partial Differential Equations
• Latent Dynamics
• Riemannian Geometry
• Out-of-Distribution Robustness

Best for: AI Scientist, Machine Learning Engineer, Research Scientist

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Editorial summary, takeaway, and curation by AIssential. Original article published by stat.ML updates on arXiv.org.