⚡ electrical engineering

KKL Observer Synthesis for Nonlinear Systems via Physics-Informed Learning

This paper proposes a novel physics-informed learning framework for designing Kazantzis-Kravaris (KKL) observers for nonlinear systems by using neural networks to learn the required state transformation and its inverse, while providing theoretical guarantees on robustness and non-asymptotic performance that demonstrate superior generalization compared to existing methods.

Original authors: M. Umar B. Niazi, John Cao, Matthieu Barreau, Karl Henrik Johansson

Published 2026-08-12

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: M. Umar B. Niazi, John Cao, Matthieu Barreau, Karl Henrik Johansson

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

The Invisible Map and the Crystal Ball

Imagine you are trying to steer a spaceship through a storm, but your cockpit only has a single, flickering lightbulb showing you the color of the clouds outside. You can't see the ship's speed, its fuel level, or its exact position. To fly safely, you need a "crystal ball"—a mathematical tool called an observer—that takes that single, fuzzy light signal and guesses the ship's full, hidden state. In the real world, sensors are often broken, expensive, or just plain missing, so engineers rely on these mathematical crystal balls to fill in the blanks.

For simple, predictable machines, we have had great crystal balls for decades. But when the machine is wild, chaotic, and non-linear (like a double pendulum swinging wildly or a heart beating irregularly), the math gets incredibly messy. One powerful method, known as the KKL observer, tries to solve this by imagining a secret, higher-dimensional "shadow world" where the chaos turns into a straight, calm line. If you can find the map to get into this shadow world and the map to get back out, you can predict the future perfectly. The problem is, finding these maps is like trying to solve a puzzle where the pieces are invisible and the picture keeps changing. This paper tackles that impossible puzzle by teaching computers to learn the maps themselves, using the laws of physics as a strict teacher.

Teaching Computers to Draw Invisible Maps

The authors of this paper, researchers from KTH Royal Institute of Technology and the University of Oxford, propose a clever new way to build these "crystal balls" for complex, non-linear systems. Their goal is to design a KKL observer, which works like a two-step magic trick. First, it takes the messy, real-world state of a system and transforms it into a higher-dimensional "shadow space" where the rules are simple and linear (like a straight line). Second, it needs to reverse the trick, taking that simple shadow and mapping it back to the real, messy world to give us an accurate guess of what's happening.

The big hurdle has always been finding these two maps: the one that goes in and the one that comes out. Traditionally, solving the math equations (partial differential equations) to find these maps is a nightmare, often impossible to do by hand. Previous attempts to use Artificial Intelligence (AI) to learn these maps often stumbled. They tried to learn the "in" map and the "out" map at the same time, like trying to learn to tie your left and right shoelaces simultaneously while blindfolded. This often confused the AI, leading to bad guesses or getting stuck in a loop where it couldn't improve.

The Paper's Solution: A Step-by-Step Dance
Instead of doing everything at once, the authors suggest a sequential learning approach. They teach the AI in two distinct stages:

Stage One (The Physics Teacher): First, they train a neural network to learn the "in" map. But here's the twist: they don't just show it data; they force it to obey the laws of physics. They use a technique called Physics-Informed Learning. Imagine teaching a student not just by showing them answers, but by giving them a textbook of rules they must follow. If the student's guess breaks a rule of physics, the teacher (the computer) gives a big penalty. This ensures the map isn't just a lucky guess based on the data, but a solution that actually respects how the system moves.
Stage Two (The Reverse Engineer): Once the "in" map is learned and locked in place, they use it to generate new, perfect practice data. Then, they train a second neural network to learn the "out" map (the inverse). Because the first map is already fixed, the second network has a clear, stable target to aim for, avoiding the confusion of the "blindfolded shoelace" problem.

What They Found
The researchers tested this method on some famous, tricky mathematical systems: the Reverse Duffing oscillator, the Van der Pol oscillator, and the chaotic Rössler and Lorenz attractors (which are famous for their butterfly-shaped chaos). They simulated these systems with noisy data, just like real-world sensors often are.

The results were promising. In their simulations, their new method (which they call SPINN) consistently guessed the hidden states more accurately than other top-tier AI methods. Crucially, it showed a superpower called generalization. When they tested the AI on situations it had never seen before—like a system swinging with much higher energy than it had ever practiced with—SPINN kept working. The other methods, which relied only on data without the "physics textbook" rules, tended to fail spectacularly outside their training zone.

The paper also provides a mathematical safety net. They proved that even if the AI makes small mistakes, the observer won't go crazy; the error will stay bounded and manageable, even if the system is noisy or slightly uncertain. They showed that the more data you give the AI and the more complex the network, the better the guess gets, but the physics rules are what keep it from going off the rails.

What It's Not
It's important to note what this paper doesn't do. The authors are careful to state that this is a simulation-based proof. They haven't built a physical robot or a real car to test this on yet; they tested it on computer models of chaotic systems. They also explicitly argue against the idea of learning the forward and backward maps simultaneously, showing that this "joint" approach often leads to worse results due to conflicting instructions for the AI. Furthermore, while they mention future work, this specific paper does not solve the problem for systems that change over time (non-autonomous systems) or systems with external controls; it focuses strictly on systems that run on their own.

In short, the paper suggests that by teaching AI to respect the laws of physics first, and then teaching it to reverse the process step-by-step, we can build much smarter, more reliable "crystal balls" for predicting the behavior of the world's most chaotic machines.

Technical Summary: KKL Observer Synthesis for Nonlinear Systems via Physics-Informed Learning

1. Problem Statement

The paper addresses the challenge of designing Kazantzis-Kravaris/Luenberger (KKL) observers for autonomous nonlinear systems. A KKL observer relies on an injective coordinate transformation map, $T$ , that lifts a nonlinear system into a higher-dimensional space where its dynamics are linear up to output injection and stable. The observer then replicates these linear dynamics and maps the state back to the original coordinates using the inverse map, $T^*$ .

The synthesis of such observers faces two primary hurdles:

Analytical Intractability: Finding the transformation map $T$ requires solving a specific partial differential equation (PDE), which is computationally difficult. Furthermore, computing the inverse map $T^*$ analytically or numerically is often infeasible.
Learning Limitations: Existing learning-based approaches (e.g., joint encoder-decoder architectures) often suffer from conflicting gradients during training, leading to suboptimal local minima and poor generalization, particularly when the system's initial state deviates from the training distribution. Additionally, purely data-driven methods may fail to respect the underlying physical constraints of the system dynamics.

2. Methodology

The authors propose a novel sequential learning framework that decouples the learning of the forward transformation map ( $T$ ) and its inverse ( $T^*$ ), utilizing Physics-Informed Neural Networks (PINNs).

2.1 Sequential Learning Architecture

Unlike previous works that train an autoencoder (jointly learning $T$ and $T^*$ ), this method employs a two-stage process:

Forward Map Learning ( $\hat{T}_\theta$ ): A neural network is trained to approximate the transformation map $T$ . The training objective combines a data-fitting term (matching synthetic trajectories) with a physics-informed regularization term. This term penalizes the residual of the governing PDE:
$\mathcal{P}_\theta(x) = \frac{\partial \hat{T}_\theta}{\partial x}(x)f(x) - A\hat{T}_\theta(x) - Bh(x)$
where $f$ and $h$ are the system dynamics and output functions, and $A, B$ are fixed observer matrices.
Inverse Map Learning ( $\hat{T}^*_\eta$ ): Once the forward map is trained and fixed, a second neural network is trained to learn the inverse. It minimizes the reconstruction risk by mapping the estimated latent states back to the original state space:
$\min_\eta \mathbb{E}_{x \sim D_X} [\|x - \hat{T}^*_\eta(\hat{T}_\theta(x))\|^2]$

2.2 Data Generation

Synthetic data is generated by simulating the system dynamics and the linear observer dynamics. To handle the unknown initial conditions of the observer state $z(0)$ , the authors leverage the stability of the observer (Hurwitz matrix $A$ ). They simulate trajectories from random initial points and discard the transient portion of the $z$ -trajectory, ensuring that the remaining data points are effectively independent of the initial observer state and lie on the invariant manifold.

2.3 Theoretical Guarantees

The paper establishes non-asymptotic learning guarantees:

Generalization Bounds: Using pseudo-dimension theory, the authors derive high-probability bounds on the approximation errors of both the forward and inverse maps.
Robustness: The paper proves that the time-average steady-state estimation error is bounded by the learning approximation error and system uncertainties (model mismatch and measurement noise). Specifically, the error converges to a neighborhood of zero determined by the quality of the learned inverse map and the magnitude of external disturbances.

3. Key Contributions

Sequential Learning Strategy: The paper introduces a sequential approach to learning KKL observers, avoiding the conflicting gradients inherent in joint encoder-decoder training. This improves optimization stability and accuracy.
Physics-Informed Regularization: By incorporating the KKL PDE as a loss term, the method enforces physical consistency, acting as a strong regularizer that prevents overfitting to training data and enables extrapolation to unseen states.
Theoretical Analysis: The work provides rigorous non-asymptotic bounds linking the empirical risk minimization to the true approximation error. It further proves that the resulting observer is robust to both learning errors and system uncertainties, guaranteeing bounded time-average estimation error.
Empirical Validation: The approach is validated on four benchmark nonlinear systems (Reverse Duffing, Van der Pol, Rössler, and Lorenz attractors), demonstrating superior generalization capabilities compared to state-of-the-art data-driven methods (Neural ODEs, deep model-free switching, and unsupervised autoencoders).

4. Results

Numerical simulations on benchmark systems demonstrate that the proposed method (referred to as SPINN) achieves:

Higher Accuracy: Lower Symmetric Mean Absolute Percentage Error (SMAPE) compared to competing methods.
Superior Generalization: The method maintains high performance when tested on initial conditions outside the training domain (out-of-domain validation), whereas purely data-driven methods (like NODE and DMF) degrade significantly.
Robustness: The observer effectively estimates states in the presence of measurement noise and model uncertainties, adhering to the theoretical bounds derived.

The authors attribute this success to the PDE constraint acting as an inductive bias, which narrows the hypothesis space and forces the neural network to learn mappings that preserve system invariants (e.g., limit cycles) even in regions not covered by the training data.

5. Significance and Claims

The paper claims that its primary significance lies in overcoming the limitations of existing learning-based observer synthesis, specifically the difficulty of finding analytical inverses and the poor generalization of purely data-driven models.

Practical Utility: The method provides a systematic way to design KKL observers for general nonlinear systems where analytical solutions are unavailable.
Reliability: By providing theoretical guarantees on robustness and non-asymptotic error bounds, the paper positions the learned observer as a reliable tool for safety-critical applications, unlike "black-box" learning approaches.
Future Directions: The authors modestly note that future work will focus on extending this framework to non-autonomous and controlled systems (which involve time-varying PDEs) and integrating system identification with observer design for scenarios with unknown system models. They also suggest exploring active learning to optimize data sampling.

The paper concludes that the proposed physics-informed, sequential learning approach offers a robust and generalizable solution for KKL observer synthesis, bridging the gap between rigorous control theory and modern machine learning.