PINN-Obs: Adaptive Neural State Observer

• PINN-Obs is a state estimation framework that couples neural network approximation with embedded physical models to monitor nonlinear dynamics.
• It learns both the state trajectories and adaptive observer gains concurrently without relying on linearization or manual tuning.
• Empirical results show the method achieves robust accuracy and reduced estimation errors in systems like the reverse Duffing oscillator and induction motor.

An Adaptive Physics-Informed Neural Network-based Observer (PINN-Obs) is a modern state estimation framework that couples neural network function approximation with embedded models of physical system dynamics for nonlinear dynamical systems. The approach directly integrates sensor data and system differential equations into a unified, adaptive learning process—eschewing explicit system linearization, coordinate transformation, or manual gain scheduling. PINN-Obs is motivated by the need for robust, accurate, and noise-tolerant state estimation in settings where only partial and noisy measurements are available and the system may exhibit pronounced nonlinearities (2507.06712).

1. Core Principles and Observer Formulation

PINN-Obs operates by constructing an observer whose internal state trajectory and gain matrix are both produced by a deep neural network, subject to physics-based constraints. The observed system is assumed to be governed by a general nonlinear ODE of the form

$$\left\{ \begin{array}{l} \dot{x}(t) = f(x(t), t) + B u(t) \ y(t) = C x(t) \ x(0) = x_0 \end{array} \right.$$

where $x(t) \in \mathbb{R}^n$ is the state, $u(t)$ an input, and $y(t) \in \mathbb{R}^q$ the measured output.

The PINN-Obs reconstructs $x(t)$ from $y(t)$ by defining a coupled observer model: $$\left\{ \begin{array}{l} \dot{\hat{x}}(t) = f(\hat{x}(t), t) + L(t)C \left(x(t) - \hat{x}(t)\right) + B u(t) \ \hat{x}(0) = \hat{x}_0 \end{array} \right.$$ Here, $L(t)$ is an observer gain that is not predetermined but is inferred adaptively, as part of the neural network’s outputs (2507.06712).

Automatic differentiation is used to compute $\dot{\hat{x}}(t)$, enabling enforcement of dynamic model consistency during optimization.

2. Loss Function and Adaptive Gain Learning

The training of the observer is formulated as minimization of a composite Mean Squared Error (MSE)-type loss, which enforces:

• Initial condition matching:

$\text{MSE}_0 = \|\hat{x}(t_0) - \hat{x}_a(t_0)\|$

• Dynamic residual enforcement:

$$\text{MSE}_g = \frac{1}{N} \sum_{i=1}^{N} \left\| \dot{\hat{x}}(t_i) - f(\hat{x}(t_i), t_i) - B u(t_i) - L(t_i)C (x(t_i) - \hat{x}(t_i)) \right\|^2$$

• Measurement matching:

$$\text{MSE}_y = \frac{1}{N} \sum_{i=1}^{N} \|C \hat{x}(t_i) - y(t_i)\|^2$$

The objective is

$$\text{MSE} = \text{MSE}_0 + \text{MSE}_g + \text{MSE}_y$$

The observer gain $L(t)$ is learned in tandem with the state estimate $\hat{x}(t)$ as trainable outputs of the PINN, thus directly adapting to the measurement and model context (2507.06712).

3. Convergence and Theoretical Guarantees

A formal convergence analysis demonstrates that, under standard conditions (local Lipschitz continuity of $f$, network smoothness in a H\"older space, and weak system observability), minimization of the loss assures uniform convergence of the state estimate and measurement residuals across the time interval:

• The initial state error $\|\hat{x}(t_0) - \hat{x}_a(t_0)\|$ vanishes as $N \to \infty$.
• The dynamic residual (observer model error) converges uniformly to zero.
• The estimated output $C\hat{x}(t)$ uniformly matches the observed $y(t)$. These guarantees are established via function space convergence arguments (e.g., Arzelà-Ascoli theorem, uniform convergence in $C^0(\Omega)$), providing theoretical assurance of consistent estimation under mild observability (2507.06712).

4. Practical Implementation and Adaptability

Workflow Summary

• Network design: The network is typically a multilayer perceptron (MLP) with input $t$ (or $[t, x_0]$), outputting both state estimate $\hat{x}(t)$ and gain $L(t)$; architectures with 9 layers and 20 neurons per hidden layer, using tanh activation, have been used for their smoothness and bounded derivative properties.
• Training: The PINN-Obs is trained offline with available measurement data using the composite loss. Stochastic gradient descent (Adam optimizer) is used for efficient convergence.
• Inference: The trained observer can be deployed for state estimation in real time, directly producing both the state trajectory and the associated observer gains as functions of $t$.

Comparative Strengths

Relative to traditional observer schemes (e.g., Luenberger observer, Kalman filter, supervised neural-Luenberger observers, and unsupervised autoencoder methods), PINN-Obs offers several advantages:

• Unified adaptive learning: Simultaneously estimates states and observer gain without system linearization or explicit model transformation.
• Model-free gain computation: The observer gain adapts automatically and continuously, improving robustness against modeling and measurement errors.
• Direct physics enforcement: Embeds the system ODEs in the training loss, reducing prediction drift and improving generalization in data-sparse or noisy regimes.

5. Empirical Validation and Performance

Extensive numerical experiments were performed on benchmark and practical nonlinear systems:

• Reverse Duffing Oscillator: PINN-Obs achieves superior accuracy, with orders-of-magnitude reduction in MSE compared to baseline methods and robust state recovery even when only one variable is measured.
• Induction Motor Model: Successful estimation of both directly and indirectly measurable states, demonstrating practical value for electrical drives.
• Satellite Motion System and Harmonic Oscillator: The observer is adaptable across domains, maintaining low estimation errors and resilience to measurement noise.
• Architectural studies: Empirical analysis confirms that deeper architectures (nine layers) yield the best trade-off between computational cost and estimation robustness.

Performance metrics include MSE, RMSE, MAE, and SMAPE, with consistent reports of PINN-Obs outperforming existing neural state estimator designs by an order of magnitude in error reduction.

6. Limitations, Challenges, and Future Directions

Significant challenges remain in scaling PINN-Obs to high-dimensional or highly stochastic systems and in responding efficiently to changes in system dynamics or noise characteristics. Open areas for future research include:

• Enhancing measurement noise robustness via probabilistic or adversarial PINN variants.
• Extending to hybrid and stochastic systems, potentially integrating process noise modeling.
• Real-time training speedups and computational optimization for high-dimensional cases.
• Automated architectural design and loss-weight adaptation for increased generality and reduced manual tuning effort.

7. Applications and Impact

PINN-Obs offers a comprehensive and theoretically grounded method for adaptive nonlinear state estimation in a variety of engineering settings, including:

• Power system dynamics and security assessment.
• Electrical machine control and monitoring.
• Aerospace and robotics systems with nonlinear, uncertain, or partially known models.
• Data-driven observers in next-generation industrial control systems.

The integration of neural network adaptation, direct physics enforcement, and gain learning positions PINN-Obs as a significant advance in the construction of robust state observers for nonlinear dynamical systems. Its model-free, adaptive capability presents a pathway to observer designs that remain accurate, reliable, and efficient in the presence of limited or noisy sensing and complex underlying dynamics (2507.06712).