Lyapunov-Based Dynamics Modeling for Globally Stable State Prediction
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Solution Overview
Problem
Existing machine learning approaches struggle to ensure global stability of dynamics models for physical systems, particularly when trained from unstructured video data, leading to unpredictable behavior in unseen states, which is critical for real-life applications like robotic arms and autonomous vehicles.
Innovation Solution
A machine learning system that jointly learns a dynamics model and a Lyapunov function, ensuring global stability by projecting nominal dynamics onto a function that fulfills the Lyapunov condition, using an input-convex neural network and orthogonal projection to guarantee stability across the entire state space.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Reliability
If a generic neural network is used to model the dynamics of a physical system, then the model can be trained on available data, but the stability properties of the learned neural network cannot be guaranteed and enormous data sets are needed to ensure correct long-term behavior
Solution Approach 1:
The patent changes the parameter space by introducing a Lyapunov function as an additional learned parameter that constrains the dynamics model. This Lyapunov function serves as a stability certificate, transforming the training process from purely data-driven to physics-constrained, thereby reducing the need for enormous training datasets while guaranteeing stability properties.
Solution Approach 2:
The Lyapunov function acts as an intermediary between the neural network dynamics model and the stability requirement. Instead of directly enforcing stability constraints on the complex neural network, the Lyapunov function serves as a mathematical mediator that certifies stability, allowing the system to achieve reliability with less training data.
2Reliability
If stability is softly enforced as an additional loss term on training data, then some stability properties can be achieved on training data, but little can be said about the stability of the learned neural network for unseen states
Solution Approach 1:
The patent applies preliminary action by constructing the dynamics model in a specific mathematical form that inherently satisfies stability conditions. The Lyapunov function is built into the model architecture before training, ensuring that stability is guaranteed a priori for all states, not just on training data. This preliminary structural constraint enables the model to generalize stability to unseen states.
3Extent of automation
If a neural network is trained to learn dynamics from video data, then the transformation from observation space to latent space can be learned, but the characteristic long-term behavior such as stability at equilibrium points cannot be built automatically
Solution Approach 1:
The patent changes the parameterization of the dynamics model by expressing it in terms of a Lyapunov function and its gradient. This parameter change transforms the automatic learning process to inherently produce stable dynamics, as the Lyapunov-based formulation automatically ensures that equilibrium points are stable and long-term behavior is physically meaningful.
Data Source
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AI summary
A system and computer-implemented method are provided for training a dynamics model to learn the dynamics of a physical system. In particular, the dynamics model may be learned to be able to infer a future state of the physical system and/or its environment based on a current state of the physical system and/or its environment. The learned dynamics model is inherently globally stable. Namely, instead of learning a dynamics model and attempting to separately verify its stability, the learnable dynamics model comprises a learnable Lyapunov function which is jointly learned together with the nominal dynamics of the physical system. Accordingly, the learned dynamics model is highly suitable for real-life applications in which a physical system may assume a state which was unseen during training as the learned dynamics model is inherently globally stable.