⚡ electrical engineering

Environment-Aware Stable Neural Koopman Dynamics Learning for Input-Driven Systems under Environmental Constraints

This paper proposes Environment-Aware Stable Neural Koopman Dynamics Learning (ESNKD), a unified framework that integrates fiber bundle encoders, input-conditioned Neural ODEs, contraction synthesis, and Koopman lifting to provide rigorous stability and input-to-state stability guarantees for nonlinear systems operating under varying environmental constraints.

Original authors: Lin Feng

Published 2026-06-15

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Lin Feng

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to teach a robot to walk. You show it a video of a person walking on a smooth floor, and the robot learns to mimic that. But then, you ask the robot to walk on a slippery icy patch or while carrying a heavy backpack. If the robot was only trained on the smooth floor, it might fall over because it doesn't understand that the "environment" has changed.

This paper introduces a new method called ESNKD (Environment-Aware Stable Neural Koopman Dynamics Learning) to solve exactly this problem. It's a way for computers to learn how physical systems move, even when the conditions around them keep changing, while also guaranteeing that the system won't go haywire.

Here is how it works, broken down into simple concepts and analogies:

1. The Problem: The "One-Size-Fits-All" Trap

Most current AI models for physics are like a chef who only knows how to cook one specific dish perfectly. If you give them a new ingredient (like a change in friction, weight, or wind), they get confused. They try to guess the new conditions based on what they see, but they often get it wrong, leading to crashes or instability.

2. The Solution: The "Four-Part Toolkit"

The authors built a system with four special tools that work together like a well-oiled machine:

Tool 1: The "Environmental Translator" (Bundle-Structured Encoder)
Imagine the robot has a pair of special glasses. Instead of just seeing the world as a blurry mess of data, these glasses translate the environment (like "icy," "heavy load," or "windy") into a clean, organized map.
- The Paper's Claim: This tool takes noisy sensor data and turns it into a precise "coordinate" that tells the robot exactly what kind of world it is in. It ensures that "icy" and "dry" are treated as distinct, separate places on the map, not just random noise.
Tool 2: The "Smart Driver" (Input-Conditioned Neural ODE)
This is the engine that actually moves the robot. Usually, engines are built to run on a fixed track. This one is special because it has a "passenger seat" for the environment.
- The Paper's Claim: It takes the "coordinates" from the translator (Tool 1) and the control inputs (like "move forward") and mixes them together. It doesn't just guess; it actively adjusts its driving style based on the current environment, much like a driver shifting gears when going up a hill.
Tool 3: The "Stability Brake" (Contraction Synthesis Layer)
This is the safety mechanism. Imagine a car that has a built-in rule: "No matter how hard you turn, you must always get closer to the center of the road, never further away."
- The Paper's Claim: During training, the system forces the math to behave like this. It uses a specific mathematical check (a "hinge penalty") to ensure that if two robots start in slightly different spots, they will eventually converge to the same path. This guarantees the system won't spiral out of control.
Tool 4: The "Safety Certificate" (Koopman Lifting & ISS Verification)
This is the final exam. Before the robot is allowed to drive, a mathematician checks its work.
- The Paper's Claim: The system translates the complex, messy movement of the robot into a simpler, linear language (like turning a complex song into a simple sheet music score). It then runs a computer test (called an LMI) to prove, with 100% mathematical certainty, that the robot is "Input-to-State Stable" (ISS). This means: "No matter how much the wind blows or how heavy the load gets, the robot will stay safe."

3. How They Tested It

The authors didn't just talk about theory; they tested this on five different "playgrounds":

A swinging pendulum with changing friction.
A cart with a pole that changes weight.
A simulated cheetah running on surfaces with different grip.
A real robot arm lifting unknown heavy weights.
Another robot arm dealing with joint friction.

The Results:

Better Accuracy: The new method made fewer mistakes in predicting where the robot would be compared to five other popular methods.
Safer: It had the lowest rate of "stability violations" (times the robot almost fell or went crazy).
Certified: It was the only method that could successfully pass the "Safety Certificate" test for almost all its runs. Other methods either failed the test or couldn't even take the test because they were too complex.

4. The Bottom Line

Think of ESNKD as teaching a robot not just how to move, but how to adapt to the world around it, while simultaneously handing the robot a "safety license" that proves it will never lose control.

The paper claims that by combining these four specific techniques, they created a system that is more accurate, more robust to changes in the environment, and mathematically guaranteed to be stable, outperforming previous methods in every category they tested.

Technical Summary: Environment-Aware Stable Neural Koopman Dynamics Learning for Input-Driven Systems under Environmental Constraints

1. Problem Statement

The paper addresses the challenge of constructing predictive models for nonlinear dynamical systems from measurement data, specifically under environment-varying operating conditions. While existing methods have advanced the field, they fail to simultaneously address three critical requirements within a unified, trainable framework:

Environment Variability: Most learned models treat the environment as fixed or rely on implicit encoding of physical parameters (e.g., payload, friction) within the state, which fails when conditions shift outside the training distribution.
Rigorous Stability Guarantees: Deep learning approaches often lack interpretability and stability certification. While Neural ODEs and Koopman operators offer pathways to stability, they are rarely combined with environmental conditioning.
Input-to-State Stability (ISS) Certification: Existing methods struggle to provide analytical ISS certificates for systems with time-varying inputs and shifting environmental parameters.

The core objective is to learn a parametric model that minimizes trajectory prediction error on unseen data while providing analytical certificates for convergence and ISS without requiring sampling.

2. Methodology: The ESNKD Framework

The proposed Environment-Aware Stable Neural Koopman Dynamics (ESNKD) framework integrates four coupled components to address the identified gaps:

A. Bundle-Structured Environmental Encoding

Inspired by the fiber bundle framework, the method employs an encoder $\psi_\phi$ that maps noisy environmental observations $y_t$ to a structured latent coordinate $z_t$ .

Geometric Regularization: Unlike standard encoders, this component includes an isometry regularizer ( $L_{enc}$ ) to preserve the metric structure of the environmental manifold. This prevents spurious coupling between distinct environmental conditions, ensuring the learned vector field remains robust to environmental perturbations.
Role: The latent coordinate $z$ parameterizes the vector field, allowing the model to adapt dynamically to changing physical parameters (e.g., friction, mass).

B. Input-Conditioned Neural Dynamics

The system dynamics are modeled as a decomposed vector field:
$\dot{x} = f_\theta(x, u, z) = f_s(x) + f_r(x, u, z)$

Structure: $f_s(x) = -Kx$ provides a linear stabilizing baseline. $f_r$ is a neural residual that incorporates state $x$ , control input $u$ , and environment $z$ .
Coupling: The environment-input coupling is realized via a Feature-wise Linear Modulation (FiLM) layer, linking environmental coordinates directly to the vector field. This extends the "input concomitant" philosophy of ICODEs to include environmental conditioning.
Stabilization: Spectral normalization is applied to constrain the Lipschitz constant of the residual, stabilizing Jacobian computations.

C. Contraction Synthesis Layer

To enforce convergence, the framework adopts a contraction analysis approach analogous to ControlSynth Neural ODEs (CSODEs).

Condition: The system is trained to satisfy a contraction condition where the symmetric part of the Jacobian is negative definite: $\text{sym}(\frac{\partial f_\theta}{\partial x}) \preceq -\beta I$ .
Training: A hinge penalty ( $L_{con}$ ) is minimized during training to ensure the maximum eigenvalue of the symmetric Jacobian remains below $-\beta$ . This is architecture-agnostic and handles the environment dependence through $z$ .

D. Koopman Lifting and ISS Verification

Following the theoretical pipeline of recent work on ISS-verified Koopman models, the framework performs a post-hoc verification step:

Lifting: An observable dictionary $\phi(x, z)$ (comprising degree-two monomials of state and environment) is constructed.
Identification: The lifted dynamics are approximated as a linear system $\dot{\phi} = A\phi + Bu$ via least-squares regression.
Certification: Input-to-State Stability (ISS) is verified by solving a Linear Matrix Inequality (LMI) feasibility problem. If feasible, this provides an analytical certificate of ISS with explicit gain bounds, converting a robustness question into a tractable convex program.

3. Theoretical Guarantees

The paper provides rigorous theoretical proofs covering:

Existence and Uniqueness: Solutions exist and are unique under standard Lipschitz conditions.
Incremental Exponential Stability: If the contraction condition holds, any two trajectories driven by the same input and environment converge exponentially at rate $\beta$ .
ISS with Explicit Bounds: If the LMI is feasible, the lifted system is ISS with a gain dependent on the Koopman matrices.
Robustness: The model is robust to environmental perturbations; the error bound is proportional to the distortion in the latent environmental manifold, justifying the isometry regularizer.

4. Experimental Results

The framework was evaluated on five benchmark systems, including damped pendulums, cart-poles, HalfCheetah (MuJoCo), and two robotic manipulation platforms (Franka Emika Panda and xArm) under unknown payloads and friction perturbations.

Prediction Accuracy: ESNKD achieved the lowest Root Mean Square Error (RMSE) across all tasks, showing relative improvements of 26.3% and 21.5% over the best baseline (ISS-Koopman) on the pendulum and HalfCheetah tasks, respectively.
Stability and Safety: ESNKD attained the lowest Stability Violation Rate (SVR) on every task (mean 2.4% vs. 7.2% for ISS-Koopman and 17.9% for Neural ODE).
ISS Certification: The method achieved an ISS Certification Success Rate (ISS-SR) of 85–94.7% across tasks. In contrast, methods without Koopman lifting (Neural ODE, ICODE, ControlSynth) could not produce certificates, and Deep Koopman achieved at most 47.8%.
Ablation Study: Removing the environmental encoder increased RMSE by 63%. Removing the contraction loss reduced ISS-SR by 46 percentage points, confirming that stability regularization is essential for safety certification.

5. Significance and Claims

The paper claims that ESNKD is the first unified framework to simultaneously combine environmental encoding, stability synthesis, and ISS certification.

Bridging Gaps: It addresses the structural gap where environmental encoding, stability synthesis, and ISS certification have previously been treated separately.
Geometric Regularity: It demonstrates that the geometric regularity of bundle-structured latent representations improves the quality of Koopman lifting and stability certification.
Practical Attainability: The results suggest that stability-certified Neural ODEs are practically attainable even under environment-varying conditions and time-varying inputs.
Safe Exploration Pathway: The authors suggest that the obtained ISS certificates enable the use of linear robust Model Predictive Control (MPC) for safe exploration, complementing measurement-aware Control Barrier Function (CBF) approaches.

The paper concludes that while the Koopman identification is currently performed post-hoc, the framework establishes a viable pathway for learning provably stable, environment-aware dynamics models from data.