pith. sign in

arxiv: 2009.08842 · v1 · submitted 2020-09-18 · 💻 cs.CR

Physics-Informed Neural Networks for Securing Water Distribution Systems

Solon Falas , Charalambos Konstantinou , Maria K. Michael This is my paper

Pith reviewed 2026-05-24 14:16 UTC · model grok-4.3

classification 💻 cs.CR
keywords physics-informed neural networkswater distribution systemscyber-physical systemsattack mitigationcontroller securitypartial differential equations
0
0 comments X

The pith

Physics-informed neural networks mitigate the effects of controller attacks on water distribution systems.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper applies physics-informed neural networks to a water distribution network where an attacker compromises a pump controller by feeding it false liquid flow sensor readings. PINNs incorporate the governing nonlinear partial differential equations of fluid dynamics directly into network training, allowing the model to learn accurate system behavior from minimal data even under attack. This produces estimates that can be used to counteract the bad measurements and restore proper pump regulation. A reader would care because the method offers a data-efficient way to add a physics-based layer of resilience to critical infrastructure controllers without requiring complete sensor trust.

Core claim

PINNs can be trained to solve the supervised task of representing water network dynamics while satisfying the nonlinear partial differential equations that describe fluid flow, thereby mitigating the impact of attacks that falsify controller inputs from flow sensors.

What carries the argument

Physics-informed neural networks that embed the physical laws given by general nonlinear partial differential equations into the training loss function.

If this is right

  • The PINN can produce corrected flow estimates that allow the controller to continue regulating the pump despite falsified sensor data.
  • Training succeeds with only a small number of data points because the embedded PDEs constrain the solution space.
  • The same framework can be tested on other cyber-physical systems whose dynamics obey known partial differential equations.
  • Implementation challenges include choosing appropriate collocation points for the PDE residual and balancing data loss against physics loss during training.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The approach could be combined with traditional anomaly detectors so that PINN outputs are used only when sensor trust falls below a threshold.
  • Online retraining of the PINN might allow it to adapt if the attack strategy or network topology changes over time.
  • Similar physics-constrained networks could be explored for state estimation in power grids or gas pipelines that share the same data-scarce, safety-critical profile.

Load-bearing premise

The physical laws expressed by general nonlinear partial differential equations remain an accurate and usable description of the water distribution system even when the controller is under attack and sensor measurements are compromised.

What would settle it

Simulate the attacked water network, feed the PINN the compromised sensor values, and check whether the resulting pump control commands keep the actual flow and pressure within safe bounds compared to an unattacked ground-truth simulation.

Figures

Figures reproduced from arXiv: 2009.08842 by Charalambos Konstantinou, Maria K. Michael, Solon Falas.

Figure 1
Figure 1. Figure 1: PINN-enhanced smart water distribution network: A liquid pump’s [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: The x-component (U) velocity field representation of the whole channel as depicted by the graphical representation tool ParaView. Liquid flows from left to right with top and bottom boundaries acting as solid wall surfaces. A small vortex is created in the channel due to the sensor and points directly behind it move slower in the x-direction. Definition 1 (Observability): A system is said to be ob￾servable… view at source ↗
Figure 4
Figure 4. Figure 4: A snapshot representation of the velocity [PITH_FULL_IMAGE:figures/full_fig_p004_4.png] view at source ↗
read the original abstract

Physics-informed neural networks (PINNs) is an emerging category of neural networks which can be trained to solve supervised learning tasks while taking into consideration given laws of physics described by general nonlinear partial differential equations. PINNs demonstrate promising characteristics such as performance and accuracy using minimal amount of data for training, utilized to accurately represent the physical properties of a system's dynamic environment. In this work, we employ the emerging paradigm of PINNs to demonstrate their potential in enhancing the security of intelligent cyberphysical systems. In particular, we present a proof-of-concept scenario using the use case of water distribution networks, which involves an attack on a controller in charge of regulating a liquid pump through liquid flow sensor measurements. PINNs are used to mitigate the effects of the attack while demonstrating the applicability and challenges of the approach.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 0 minor

Summary. The manuscript presents a proof-of-concept application of Physics-Informed Neural Networks (PINNs) to enhance security in cyber-physical systems, specifically water distribution networks. It describes using PINNs to mitigate controller attacks on a liquid pump regulated by flow sensor measurements by enforcing physical laws expressed as general nonlinear partial differential equations.

Significance. If the results hold, the work illustrates a potential route for incorporating domain physics into neural networks to achieve attack mitigation in CPS settings with limited or corrupted data, leveraging the invariance of the underlying fluid equations under cyber compromise.

major comments (1)
  1. [Abstract] Abstract: The abstract states the intended use of PINNs for attack mitigation but supplies no quantitative results, error metrics, baseline comparisons, or implementation details, so the available text does not allow verification that any data or derivation supports the claim.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the constructive feedback on our proof-of-concept manuscript. We address the single major comment below.

read point-by-point responses

  1. Referee: [Abstract] Abstract: The abstract states the intended use of PINNs for attack mitigation but supplies no quantitative results, error metrics, baseline comparisons, or implementation details, so the available text does not allow verification that any data or derivation supports the claim.

    Authors: We agree that the abstract would be strengthened by summarizing key quantitative findings from the experiments. The manuscript body reports mitigation performance under the described attack scenario along with relevant error metrics and implementation choices. In the revised version we will expand the abstract to include these highlights while preserving its concise, high-level character. revision: yes

Circularity Check

0 steps flagged

No significant circularity; application relies on external PDEs

full rationale

The paper describes a proof-of-concept use of PINNs to mitigate controller attacks on a water-distribution pump. PINNs are introduced as an established technique that embeds independent physical laws (nonlinear PDEs for fluid flow) into the training loss; these PDEs are not derived from, fitted to, or redefined by the attack-mitigation results. No self-definitional equations, predictions that reduce to fitted inputs, or load-bearing self-citations appear in the provided text. The central claim therefore rests on the external validity of the hydraulic model rather than any internal reduction to the paper's own data or prior results.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The abstract invokes the standard definition of PINNs but supplies no explicit free parameters, invented entities, or additional axioms beyond the general claim that nonlinear PDEs describe the system physics.

axioms (1)
  • domain assumption General nonlinear partial differential equations describe the physical laws of the water distribution system.
    Stated in the opening sentence of the abstract as the foundation for PINN training.

pith-pipeline@v0.9.0 · 5663 in / 1285 out tokens · 32736 ms · 2026-05-24T14:16:11.313258+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

16 extracted references · 16 canonical work pages · 1 internal anchor

  1. [1]

    Smart water networks and cyber security,

    A. Rasekh et al., “Smart water networks and cyber security,” American Society of Civil Engineers , 2016

    work page 2016

  2. [2]

    Two more cyber-attacks hit Israel’s water system,

    C. Cimpanu, “Two more cyber-attacks hit Israel’s water system,” Jul 2020. [Online]. Available: https://www.zdnet.com/article/ two-more-cyber-attacks-hit-israels-water-system/

    work page 2020

  3. [3]

    Driven by data or derived through physics? a review of hybrid physics guided machine learning techniques with cyber- physical system (cps) focus,

    R. Rai and C. K. Sahu, “Driven by data or derived through physics? a review of hybrid physics guided machine learning techniques with cyber- physical system (cps) focus,” IEEE Access , vol. 8, pp. 71 050–71 073, 2020

    work page 2020

  4. [4]

    Enhanced resilient state estimation using data-driven auxiliary models,

    O. M. Anubi and C. Konstantinou, “Enhanced resilient state estimation using data-driven auxiliary models,” IEEE Transactions on Industrial Informatics, vol. 16, no. 1, pp. 639–647, 2020

    work page 2020

  5. [5]

    Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations,

    M. Raissi, P. Perdikaris, and G. E. Karniadakis, “Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations,” Journal of Computational Physics , vol. 378, pp. 686–707, 2019

    work page 2019

  6. [6]

    Finite element approximation of the Navier-Stokes equations,

    V . Girault and P.-A. Raviart, “Finite element approximation of the Navier-Stokes equations,” LNM, vol. 749, 1979

    work page 1979

  7. [7]

    False data injection attacks against state estimation in wireless sensor networks,

    Y . Mo et al. , “False data injection attacks against state estimation in wireless sensor networks,” in 49th IEEE Conference on Decision and Control (CDC). IEEE, 2010, pp. 5967–5972

    work page 2010

  8. [8]

    A data-based detection method against false data injection attacks,

    C. Konstantinou and M. Maniatakos, “A data-based detection method against false data injection attacks,” IEEE Design Test, 2019

    work page 2019

  9. [9]

    Adversarial examples on power systems state estimation,

    A. Sayghe, O. M. Anubi, and C. Konstantinou, “Adversarial examples on power systems state estimation,” in IEEE PES Innovative Smart Grid Technologies Conference (ISGT). IEEE, 2020, pp. 1–5

    work page 2020

  10. [10]

    Model-based attack detection scheme for smart water distribution networks,

    C. M. Ahmed, C. Murguia, and J. Ruths, “Model-based attack detection scheme for smart water distribution networks,” in Proceedings of the 2017 ACM ASIACCS , 2017, pp. 101–113

    work page 2017

  11. [11]

    A secure hybrid dynamic state estimation approach for power systems under false data injection attacks,

    Z. Kazemi et al. , “A secure hybrid dynamic state estimation approach for power systems under false data injection attacks,” IEEE Transactions on Industrial Informatics , 2020

    work page 2020

  12. [12]

    Secure state estimation for cyber-physical systems under sensor attacks: A satisfiability modulo theory approach,

    Y . Shoukry et al. , “Secure state estimation for cyber-physical systems under sensor attacks: A satisfiability modulo theory approach,” IEEE Transactions on Automatic Control , vol. 62, no. 10, pp. 4917–4932, 2017

    work page 2017

  13. [13]

    Physics Informed Deep Learning (Part II): Data-driven Discovery of Nonlinear Partial Differential Equations

    M. Raissi, P. Perdikaris, and G. E. Karniadakis, “Physics informed deep learning (part ii): Data-driven discovery of nonlinear partial differential equations,” arXiv preprint arXiv:1711.10566 , 2017

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  14. [14]

    Gmsh: A 3-d finite element mesh generator with built-in pre-and post-processing facilities,

    C. Geuzaine and J.-F. Remacle, “Gmsh: A 3-d finite element mesh generator with built-in pre-and post-processing facilities,” International journal for numerical methods in engineering, vol. 79, no. 11, pp. 1309– 1331, 2009

    work page 2009

  15. [15]

    Nektar++: An open-source spectral/hp element framework,

    C. D. Cantwell et al. , “Nektar++: An open-source spectral/hp element framework,” Computer physics communications , vol. 192, pp. 205–219, 2015

    work page 2015

  16. [16]

    maziarraissi/pinns

    M. Raissi, “maziarraissi/pinns.” [Online]. Available: https://github.com/ maziarraissi/PINNs

    work page