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Interpretable Meta-Learning of Physical Systems

Matthieu Blanke, Marc Lelarge

TL;DR

It is argued that multi-environment generalization can be achieved using a simpler learning model, with an affine structure with respect to the learning task, and it is proved that this architecture can identify the physical parameters of the system, enabling interpreable learning.

Abstract

Machine learning methods can be a valuable aid in the scientific process, but they need to face challenging settings where data come from inhomogeneous experimental conditions. Recent meta-learning methods have made significant progress in multi-task learning, but they rely on black-box neural networks, resulting in high computational costs and limited interpretability. Leveraging the structure of the learning problem, we argue that multi-environment generalization can be achieved using a simpler learning model, with an affine structure with respect to the learning task. Crucially, we prove that this architecture can identify the physical parameters of the system, enabling interpreable learning. We demonstrate the competitive generalization performance and the low computational cost of our method by comparing it to state-of-the-art algorithms on physical systems, ranging from toy models to complex, non-analytical systems. The interpretability of our method is illustrated with original applications to physical-parameter-induced adaptation and to adaptive control.

Interpretable Meta-Learning of Physical Systems

TL;DR

It is argued that multi-environment generalization can be achieved using a simpler learning model, with an affine structure with respect to the learning task, and it is proved that this architecture can identify the physical parameters of the system, enabling interpreable learning.

Abstract

Machine learning methods can be a valuable aid in the scientific process, but they need to face challenging settings where data come from inhomogeneous experimental conditions. Recent meta-learning methods have made significant progress in multi-task learning, but they rely on black-box neural networks, resulting in high computational costs and limited interpretability. Leveraging the structure of the learning problem, we argue that multi-environment generalization can be achieved using a simpler learning model, with an affine structure with respect to the learning task. Crucially, we prove that this architecture can identify the physical parameters of the system, enabling interpreable learning. We demonstrate the competitive generalization performance and the low computational cost of our method by comparing it to state-of-the-art algorithms on physical systems, ranging from toy models to complex, non-analytical systems. The interpretability of our method is illustrated with original applications to physical-parameter-induced adaptation and to adaptive control.
Paper Structure (51 sections, 2 theorems, 16 equations, 8 figures, 3 tables, 2 algorithms)

This paper contains 51 sections, 2 theorems, 16 equations, 8 figures, 3 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Contributions
  3. Learning from multiple physical environments
  4. The variability of physical systems
  5. Overview of multi-environment deep learning
  6. Learning model
  7. Meta-training
  8. Test-time adaptation
  9. Computational cost
  10. Context-Affine Multi-Environment Learning
  11. Problem structure
  12. Computational benefits
  13. Applicability
  14. Interpretability and system identification
  15. Physical context identification
  16. ...and 36 more sections

Key Result

Proposition 1

Assume that the training points are uniform across tasks: $x_{t}^{(i)} = x^{(i)}$, and ${N_t=N}$ for all $1 \leq t \leq T$ and $1 \leq i \leq N$, with $n\leq r<N, T$. Assume that both sets $\{ \nu(x^{(i)}) \}$ and $\{ \varphi_t \}$ span $\mathbb{R}^{n}$. In the limit of a vanishing training loss $L

Figures (8)

  • Figure 1: Few-shot adaptation on two out-of-domain environments of the point charge system in a dipolar setting (left) and the capacitor (right). The adaptation points are represented by the $\times$ symbols. The vector fields are derived from the learned potential fields using automatic differentiation.
  • Figure 2: Average relative error for the point charge identification.
  • Figure 3: Upkie.
  • Figure 4: Tracking of a reference trajectory using the learned inverse dynamics controller. Left. 50-shot adaptation. Center and right. The model and the controller are adapted online.
  • Figure 5: Adaptation and relative identification error for the $\varepsilon$-capacitor, with increasing $\varepsilon$.
  • ...and 3 more figures

Theorems & Definitions (10)

  • Example 1: Actuated pendulum
  • Example 2: Electrostatic potential
  • Definition 1: Context-affine multi-task learning
  • Example 3: Electric point charges
  • Example 4: Inverse dynamics in robotics
  • Example 5: Identification of boundary perturbations
  • Proposition 1
  • Lemma 1
  • proof : Proof of Lemma \ref{['lemma:symmetry']}
  • proof : Proof of Proposition \ref{['proposition:identification']}