Physics-Informed Neural Networks (PINNs)

• Physics-Informed Neural Networks (PINNs) are deep learning models that embed governing PDEs into the training process to enforce physical consistency.
• They use a composite loss function that combines data misfit with physics residuals, achieving high accuracy in applications like fluid dynamics.
• PINNs act as virtual sensors in cyber-physical systems, providing robust state estimation even with sparse or compromised sensor data.

Physics-Informed Neural Networks (PINNs) are a data-efficient deep learning paradigm in which the governing laws of physics, often represented by partial differential equations (PDEs), are embedded directly into the neural network training process. PINNs enforce physical consistency on the model's predictions by minimizing the residuals of these governing equations, which enables them to function robustly even with sparse, unreliable, or adversarially perturbed data. PINNs have been demonstrated across a range of engineering and physics domains, with particular effectiveness in forward and inverse problems governed by complex physical laws.

1. Mathematical and Algorithmic Foundations

PINNs minimize a composite loss function integrating both data misfit and physics residuals. The essential structure is as follows:

• The neural network approximates the solution $[u(x, y, t), v(x, y, t), p(x, y, t)]$ to a physical problem.
• The loss function includes terms for both the discrepancy with observed/measured data and the residuals of the governing PDEs, evaluated at collocation points:

$$L_{\text{total}} = L_{\text{data}} + \lambda_p L_{\text{Physics}},$$

where $L_{\text{Physics}}$ includes the squared residuals of all relevant equations evaluated either analytically (via automatic differentiation) or at discrete points.

For instance, in fluid dynamics applications, the 2D Navier–Stokes equations are embedded in the PINN loss:

• Continuity: $$\frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} = 0$$
• Momentum-x: $$\frac{\partial u}{\partial t} + u\frac{\partial u}{\partial x} + v\frac{\partial u}{\partial y} = -\frac{1}{\rho} \frac{\partial p}{\partial x} + \nu\left( \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} \right)$$
• Momentum-y: $$\frac{\partial v}{\partial t} + u\frac{\partial v}{\partial x} + v\frac{\partial v}{\partial y} = -\frac{1}{\rho} \frac{\partial p}{\partial y} + \nu\left( \frac{\partial^2 v}{\partial x^2} + \frac{\partial^2 v}{\partial y^2} \right)$$

The effect is to treat the system of equations as soft constraints: while the PINN may not exactly satisfy the PDE at every point, the training is guided to keep the solutions close to the physical manifold.

2. PINNs for Security and State Estimation in Water Distribution Networks

In the domain of cyber-physical system (CPS) security for water distribution systems, PINNs function as "virtual sensors." The monitoring and detection unit (MDU) flags a sensor as potentially compromised (due to attack or failure), and the PINN is queried using the spatial coordinates and current timestamp corresponding to the missing or corrupted measurement.

Instead of extrapolating from data alone, the PINN leverages its Navier–Stokes-based physical loss constraints to predict the velocity components $\mathbf{U}$ and $\mathbf{V}$ at the queried point. This maintains system observability even under data-poisoning conditions. The reported experimental accuracy of this approach was approximately $99.45\%$ for the $U$ velocity component and $95.71\%$ for $V$ in simulation, using only minimal training data.

Distinctive strengths in this context:

• Physics Integration lessens the reliance on dense sensor coverage.
• Generalization extends the model's reliable predictions even outside the training regime.
• Attack Robustness arises because physical laws, not merely trends in recorded data, regularize the inference.
• Rapid Inference enables real-time deployment as a cyber-physical defense mechanism.

3. Comparison with Traditional State Estimation Methods

Traditional CPS state estimation often uses either:

• Physically-based numerical solvers that are computationally expensive and unsuitable for real-time recovery; or
• Purely data-driven estimators, which require dense (and trusted) measurements and are prone to manipulation.

PINNs, by integrating the physics model as a loss regularizer, require much less data to achieve high-accuracy state inference. Their surrogate sensor capability extends naturally to redundancy and resiliency strategies in cyber-physical infrastructure.

Method Type	Data Efficiency	Robustness to Attack	Real-Time Potential
Numerical Solvers	Low	High (if all input data is valid)	No
Pure ML Estimators	Low	Low	Yes
PINN-based Approach	High	High	Yes

By providing both high accuracy and computational efficiency, PINNs bridge the gap between data-centric ML and computational physics models, especially as surrogate sensors during adversarial conditions.

4. Key Implementation Details

The PINN in this context utilizes a multi-layer neural network, typically trained by minimizing the combined data-plus-physics residual loss over collocation points, sensor locations, and boundary/initial samples. Derivatives required by the PDE residuals are computed using automatic differentiation. This ensures that the optimization landscape reflects the curvature and physical structure of the underlying equations.

• Input to the PINN: spatial coordinates $(x, y)$, time $t$, system parameters.
• Output: predicted field variables (e.g., $u(x, y, t)$, $v(x, y, t)$ velocities).
• Training data: includes both direct velocity measurements and indirect enforcement of the physics via the PDE terms.
• Deployment scenario: at runtime, the PINN is invoked for locations/timepoints where the sensor data is missing or flagged; the prediction is then used for control and estimation in place of the faulty/missing measurement.

Reported results demonstrate that, after initial PINN training, inference is rapid ($<$ms in most practical cases), enabling use in closed-loop control where latency is critical.

5. Limitations and Future Directions

The main challenges highlighted are:

• Duration of PINN accuracy under sustained, evolving attack scenarios—quantifying how "stale" the PINN's model becomes as system dynamics diverge from the training scenario.
• Sensor attack/fault discrimination: the system assumes a monitoring layer exists that can reliably separate cyberattacks from classical sensor faults, which remains non-trivial.
• Validation with real-world data: current results are based on simulation; transfer to operational water networks requires field trials.
• Sequential anomaly detection: integrating PINN-generated virtual measurements as a baseline for detecting anomalous patterns in remaining sensors.

Potential future developments include hybrid anomaly detection frameworks, automated retraining pipelines for PINNs triggered by operational drift, and extensions to more complex flow regimes or other classes of cyberphysical infrastructure.

6. Broader Impact and Generalization

The successful application of PINNs as cyber-resilient surrogates in smart water distribution networks suggests significant broader impact. PINNs may be deployed in any domain combining partial physical knowledge with unreliable or expensive-to-acquire measurements, especially where adversarial intervention or natural failures are of concern. Their ability to generate physically plausible surrogate data with negligible online computational overhead positions them as a key component in modern resilient CPS architectures (Falas et al., 2020).