pith. sign in

arxiv: 2605.19826 · v1 · pith:RLPVIX2Onew · submitted 2026-05-19 · 💻 cs.AI

Explainable Wastewater Digital Twins: Adaptive Context-Conditioned Structured Simulators with Self-Falsifying Decision Support

Gary Simethy , Daniel Ortiz Arroyo , Petar Durdevic This is my paper

Pith reviewed 2026-05-20 05:57 UTC · model grok-4.3

classification 💻 cs.AI
keywords digital twinwastewater treatmentconformal risk controlexplainable AIstate-space modelsaeration controlN2O emissionsdecision support
0
0 comments X

The pith

A context-conditioned structured simulator with conformal risk control certifies safe aeration decisions in wastewater plants.

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

The paper introduces CCSS-IX, an explainable digital twin for wastewater treatment that uses a mixture of locally linear state-space models adaptively selected by context to simulate plant dynamics. It layers on a decision support system that uses conformal risk control to either approve, reject, or provide a falsifying temporal witness for proposed operator actions that lack statistical safety certification. A sympathetic reader would care if this approach allows balancing energy efficiency and environmental safety with verifiable guarantees, rather than opaque models or conservative manual rules. Tests on full-scale Danish plants and the BSM2 benchmark show the simulator stays within 1 percent error of black-box alternatives while the decision layer cuts regret and blocks many unsafe approvals.

Core claim

The authors claim that their CCSS-IX simulator, consisting of interpretable locally linear state-space experts mixed adaptively via a context-aware gating network on a continuous-time regime-switching scaffold, together with a conformal risk control decision layer that abstains or returns falsifying witnesses, supplies an end-to-end pipeline with finite-sample coverage guarantees for safe control interventions in wastewater aeration and dosing.

What carries the argument

CCSS-IX, the bank of interpretable locally linear state-space experts adaptively mixed by a context-aware gating network on a regime-switching scaffold, which carries the interpretable simulation while the conformal layer handles the certified decision support.

Load-bearing premise

The conformal risk control layer supplies valid finite-sample coverage guarantees when applied to the observed time-series with 42.6 percent sensor missingness and 2-minute sampling.

What would settle it

Running the full pipeline on a new hold-out slice from the Avedøre plant and checking whether the reported 43.6 percent regret reduction and zero unsafe actions persist under the same unsafe-action cost weight.

Figures

Figures reproduced from arXiv: 2605.19826 by Daniel Ortiz Arroyo, Gary Simethy, Petar Durdevic.

Figure 1
Figure 1. Figure 1: System overview. Plant inputs (observations, planned controls, disturbances, operating context) feed a sticky regime router; for each regime [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Risk–coverage view of the validity layer on the two-plant shadow-mode replay. Support-only abstention (blue) trades coverage for risk along a curve [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Channel-effect strength per regime at Agtrup (state / control / disturbance, left to right). Each cell is the total contribution of one variable’s channel (Ak(γ), Bk(γ), Ek(γ)) under the held-out evaluation window; rows are regimes 0–2. The numerical values are reported in the supplementary material as per-channel effect tables. 9 [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Top dependency edges per regime ranked by [PITH_FULL_IMAGE:figures/full_fig_p010_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Representative N2O spike: 2024-04-27 08:32, peak at horizon step 268, regime 2 dominant. Per-channel magnitude contribution at the peak (sum of absolute driver contributions within each channel, in raw ∆z units, not normalised to a share over channels): state 0.64, control 0.39, disturbance 0.21, additive 0.04, residual 0.00. Named top drivers: N2O (state, persistence), DO setpoint (control, low-DO formati… view at source ↗
Figure 6
Figure 6. Figure 6: Energy–N2O frontier for 27 low-DO BSM2 scenarios. Safe low-DO cases (green) cluster at median aeration-energy saving 2.41%. Unsafe low-DO cases (red) extend to 7.02% median saving. Filled circles are raw false-safe approvals: the learned rollout marks them safe but the BSM2 oracle marks them unsafe. The validity layer screens the most attractive energy-saving region, not merely unusual scenarios. 13 [PITH… view at source ↗
Figure 7
Figure 7. Figure 7: Representative temporal witness for a single low-DO intervention scenario (BSM2 scenario library identifier low-DO strategy 04). The raw full rollout [PITH_FULL_IMAGE:figures/full_fig_p014_7.png] view at source ↗
read the original abstract

Operators of safety-critical industrial processes increasingly rely on digital twins to screen control interventions, but such simulators rarely carry certified safety guarantees. Wastewater treatment plants exemplify the gap: operators face a daily safety-efficiency trade-off where aerating too little risks effluent violations and nitrous-oxide (N2O) spikes, and aerating too much wastes energy. We develop an explainable digital twin for aeration and dosing setpoints. CCSS-IX, the simulator, is a bank of interpretable locally linear state-space "experts" adaptively mixed by a context-aware gating network, building on a continuous-time regime-switching scaffold. A runtime decision layer applies conformal risk control to abstain, reopen, or return a falsifying temporal witness for any operator-proposed action that cannot be statistically certified. The artificial-intelligence contribution is twofold: an identifiable, context-conditioned structured surrogate that retains operator-readable dynamics, and a self-falsifying decision rule with finite-sample coverage guarantees. The engineering contribution is a validated, end-to-end decision-support pipeline, tested on a 1000-step slice of the Aved{\o}re full-scale plant (42.6% sensor missingness, 2-minute sampling), the Agtrup/BlueKolding full-scale plant in Denmark, and the Benchmark Simulation Model No. 2 (BSM2) international benchmark, under a matched ten-seed protocol. The static structured ensemble lies within 0.78% root-mean-square error of an unconstrained black-box reference, and the adaptive variant within 1.08%. The calibrated reopen rule cuts aggregate two-plant regret by 43.6% at an unsafe-action cost weight of 4 and eliminates unsafe chosen actions on the BSM2 main slice. Event-aligned temporal witnesses prevent 93 of 187 false-safe N2O approvals, about 4.65x the dyadic baseline (paired McNemar p < 1e-21).

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 / 2 minor

Summary. The paper introduces CCSS-IX, an explainable digital twin for wastewater aeration and dosing control. It consists of a bank of interpretable locally linear state-space experts adaptively mixed by a context-aware gating network on a continuous-time regime-switching scaffold, paired with a runtime conformal risk control layer that abstains, reopens, or returns falsifying temporal witnesses for uncertified operator actions. The system is evaluated on Avedøre (42.6% missingness, 2-min sampling) and Agtrup full-scale plants plus the BSM2 benchmark under a matched ten-seed protocol, reporting 0.78% and 1.08% RMSE relative to black-box baselines, 43.6% aggregate regret reduction at unsafe-action cost weight 4, elimination of unsafe actions on BSM2, and 93/187 prevented false-safe N2O approvals (4.65× dyadic baseline, McNemar p < 1e-21).

Significance. If the finite-sample coverage guarantees are shown to hold, the work supplies a rare combination of operator-readable structured dynamics, adaptive context conditioning, and certified self-falsification for safety-critical industrial decision support. The use of independent full-scale recordings and the public BSM2 benchmark, together with the matched multi-seed protocol and concrete regret and event-level metrics, strengthens the empirical contribution over purely synthetic or single-plant studies.

major comments (1)
  1. [Abstract / conformal risk control section] Abstract and the conformal risk control section: the central safety claim is that the runtime layer supplies finite-sample coverage for abstain/reopen decisions and temporal witnesses. Standard split conformal prediction relies on exchangeability between calibration and test points, yet the data are strongly autocorrelated 2-minute time series with 42.6% sensor missingness that must be imputed. No explicit adaptation (blocking, martingale, or time-series conformal variant) or proof that imputation preserves the required exchangeability is visible; if the proof remains the vanilla one, the nominal coverage does not transfer to the observed regime and undermines the self-falsifying guarantee.
minor comments (2)
  1. [Abstract] The RMSE values are reported as percentages relative to an unconstrained black-box reference; absolute error scales or units (e.g., mg/L for N2O or kWh for energy) would improve interpretability for operators.
  2. [Model description] The gating-network mixing weights are listed as the only free parameters; confirm that all other parameters in the locally linear experts are either fixed from first principles or identified in a fully specified procedure.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the positive overall assessment and for the constructive comment on the conformal risk control layer. We address the concern point by point below and will revise the manuscript to strengthen the theoretical and methodological presentation of the finite-sample guarantees.

read point-by-point responses

  1. Referee: [Abstract / conformal risk control section] Abstract and the conformal risk control section: the central safety claim is that the runtime layer supplies finite-sample coverage for abstain/reopen decisions and temporal witnesses. Standard split conformal prediction relies on exchangeability between calibration and test points, yet the data are strongly autocorrelated 2-minute time series with 42.6% sensor missingness that must be imputed. No explicit adaptation (blocking, martingale, or time-series conformal variant) or proof that imputation preserves the required exchangeability is visible; if the proof remains the vanilla one, the nominal coverage does not transfer to the observed regime and undermines the self-falsifying guarantee.

    Authors: We agree that the manuscript does not currently make the adaptation for temporal dependence and imputation explicit enough. Our implementation employs a blocked split-conformal procedure with block length chosen from the empirical autocorrelation function of the 2-minute series (approximately 15–20 steps) together with a forward-fill imputation that preserves the required conditional exchangeability within blocks. In the revised version we will (1) add a dedicated subsection detailing the blocking scheme and the imputation operator, (2) include a short proof sketch establishing that the coverage guarantee transfers under the observed dependence (citing standard results on conformal prediction for weakly dependent processes), and (3) report an empirical coverage check on temporally held-out folds from both full-scale plants. These additions will be placed in the conformal risk control section and referenced from the abstract; the empirical results and regret numbers remain unchanged. revision: yes

Circularity Check

0 steps flagged

No derivation circularity; performance metrics and coverage claims rest on external benchmarks rather than self-referential fits

full rationale

The reported results (43.6% regret reduction, 93 prevented false approvals, RMSE within 1.08%) are obtained from validation on independent full-scale recordings (Avedøre, Agtrup/BlueKolding) and the public BSM2 benchmark under a ten-seed protocol. No equation in the abstract or described pipeline reduces these figures to quantities defined by the model's own fitted parameters. The conformal risk control layer is invoked for finite-sample coverage, but the manuscript does not present a self-definitional reduction or load-bearing self-citation that forces the safety guarantees by construction. The central claims therefore retain independent empirical content against external data.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

Abstract-only review limits visibility into exact parameter counts and background lemmas; the central claim rests on standard conformal-prediction coverage plus the modeling assumption that locally linear experts plus context gating can approximate plant dynamics without post-hoc data exclusion.

free parameters (1)
  • gating-network mixing weights
    Context-aware gating parameters that adaptively combine the locally linear experts are fitted to training trajectories.
axioms (1)
  • domain assumption Conformal risk control supplies finite-sample coverage guarantees for the abstain/reopen decision rule even under sensor missingness and temporal correlation.
    Invoked for the runtime decision layer that certifies or falsifies operator-proposed actions.

pith-pipeline@v0.9.0 · 5896 in / 1534 out tokens · 62553 ms · 2026-05-20T05:57:06.762473+00:00 · methodology

discussion (0)

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

Reference graph

Works this paper leans on

58 extracted references · 58 canonical work pages · 7 internal anchors

  1. [2]

    A.-J. Wang, H. Li, Z. He, Y . Tao, H. Wang, M. Yang, D. Savic, G. T. Daigger, N. Ren, Digital twins for wastewater treatment: A technical review, Engineering 36 (2024) 21–35. doi:10.1016/j.eng.2024.04. 012

    work page doi:10.1016/j.eng.2024.04 2024

  2. [3]

    Rasheed, O

    A. Rasheed, O. San, T. Kvamsdal, Digital twin: Values, challenges and enablers from a modeling perspective, IEEE Access 8 (2020) 21980– 22012. doi:10.1109/ACCESS.2020.2970143

    work page doi:10.1109/access.2020.2970143 2020

  3. [4]

    F. Tao, M. Zhang, Y . Liu, A. Y . C. Nee, Digital twin driven prognostics and health management for complex equipment, CIRP Annals 67 (2018) 169–172. doi:10.1016/j.cirp.2018.04.055

    work page doi:10.1016/j.cirp.2018.04.055 2018

  4. [5]

    Ghorbani Bam, N

    P. Ghorbani Bam, N. Rezaei, A. Roubanis, D. Austin, E. Austin, B. Tar- roja, I. Takacs, K. Villez, D. Rosso, Digital twin applications in the water sector: A review, Water (MDPI) 17 (2025) 2957. doi:10.3390/ w17202957

    work page 2025

  5. [6]

    R. T. Q. Chen, Y . Rubanova, J. Bettencourt, D. K. Duvenaud, Neural ordinary differential equations, in: Advances in Neural Information Pro- cessing Systems, volume 31, 2018

    work page 2018

  6. [7]

    Kidger, J

    P. Kidger, J. Morrill, J. Foster, T. Lyons, Neural controlled differential equations for irregular time series, in: Advances in Neural Information Processing Systems, volume 33, 2020

    work page 2020

  7. [8]

    Seshan, J

    S. Seshan, J. Poinapen, M. H. Zandvoort, J. B. van Lier, Z. Kapelan, Forecasting nitrous oxide emissions from a full-scale wastewater treat- ment plant using LSTM-based deep learning models, Water Research 268 (2025) 122754. doi:10.1016/j.watres.2024.122754

    work page doi:10.1016/j.watres.2024.122754 2025

  8. [9]

    Huang, Y

    Z. Huang, Y . Bai, H. Liu, Symmetry-inspired prediction of nitrous oxide emissions in wastewater treatment using deep learning and ex- plainable analysis, Symmetry (MDPI) 17 (2025) 297. doi:10.3390/ sym17020297

    work page 2025

  9. [10]

    A. R. Ravishankara, J. S. Daniel, R. W. Portmann, Nitrous oxide (N 2O): The dominant ozone-depleting substance emitted in the 21st century, Sci- ence 326 (2009) 123–125. doi:10.1126/science.1176985

    work page doi:10.1126/science.1176985 2009

  10. [12]

    Stop explaining black box machine learning models for high stakes decisions and use interpretable models instead

    C. Rudin, Stop explaining black box machine learning models for high stakes decisions and use interpretable models instead, Nature Machine Intelligence 1 (2019) 206–215. doi:10.1038/s42256-019-0048-x

    work page doi:10.1038/s42256-019-0048-x 2019

  11. [13]

    URL:https://www.iso

    ISO/IEC, ISO/IEC TR 5469:2024 Artificial intelligence — Functional safety and AI systems, Technical Report, International Organization for Standardization, Geneva, Switzerland, 2024. URL:https://www.iso. org/standard/81283.html

    work page 2024

  12. [14]

    Geifman, R

    Y . Geifman, R. El-Yaniv, Selective classification for deep neural net- works, in: Advances in Neural Information Processing Systems, vol- ume 30, 2017

    work page 2017

  13. [15]

    A. N. Angelopoulos, S. Bates, A. Fisch, L. Lei, T. Schuster, Conformal risk control, arXiv preprint arXiv:2208.02814 (2022). doi:10.48550/ arXiv.2208.02814

    work page arXiv 2022

  14. [16]

    L. D. Hansen, A. Rani, M. A. Stokholm-Bjerregaard, P. A. Stentoft, D. Ortiz-Arroyo, P. Durdevic, Time series dataset for modeling and fore- casting of N2O in wastewater treatment, arXiv preprint arXiv:2407.05959 (2024). doi:10.48550/arXiv.2407.05959

    work page doi:10.48550/arxiv.2407.05959 2024

  15. [18]

    Jeppsson, M.-N

    U. Jeppsson, M.-N. Pons, I. Nopens, J. Alex, J. B. Copp, K. V . Gernaey, C. Rosen, J.-P. Steyer, P. A. Vanrolleghem, Benchmark simulation model no. 2: general protocol and exploratory case studies, Water Science and Technology 56 (2007) 67–78. doi:10.2166/wst.2007.604

    work page doi:10.2166/wst.2007.604 2007

  16. [19]

    Henze, W

    M. Henze, W. Gujer, T. Mino, M. C. M. van Loosdrecht, Activated Sludge Models ASM1, ASM2, ASM2d and ASM3, IW A Scientific and Technical Report 9, IW A Publishing, 2000

    work page 2000

  17. [20]

    Ljung, System Identification: Theory for the User, 2nd ed., Prentice Hall, Upper Saddle River, NJ, 1999

    L. Ljung, System Identification: Theory for the User, 2nd ed., Prentice Hall, Upper Saddle River, NJ, 1999

    work page 1999

  18. [21]

    D. E. Seborg, T. F. Edgar, D. A. Mellichamp, F. J. Doyle, Process Dy- namics and Control, 4th ed., Wiley, 2017

    work page 2017

  19. [22]

    B. Lim, S. Ö. Arık, N. Loeff, T. Pfister, Temporal fusion transformers for interpretable multi-horizon time series forecasting, volume 37, 2021, pp. 1748–1764. doi:10.1016/j.ijforecast.2021.03.012

    work page doi:10.1016/j.ijforecast.2021.03.012 2021

  20. [23]

    S. W. Linderman, M. J. Johnson, A. C. Miller, R. P. Adams, D. M. Blei, L. Paninski, Bayesian learning and inference in recurrent switching linear dynamical systems, in: Artificial Intelligence and Statistics, 2017, pp. 914–922

    work page 2017

  21. [24]

    End-to-End Identifiable and Consistent Recurrent Switching Dynamical Systems

    C. Balsells-Rodas, Z. Xiang, X. Sumba, Y . Li, End-to-end identifiable and consistent recurrent switching dynamical systems, arXiv preprint arXiv:2605.06315 (2026). doi:10.48550/arXiv.2605.06315

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.2605.06315 2026

  22. [25]

    Zhang, Y

    C. Zhang, Y . Xie, Identifiable representation and model learning for la- tent dynamic systems, arXiv preprint arXiv:2410.17882 (2024). doi:10. 48550/arXiv.2410.17882

    work page arXiv 2024

  23. [26]

    Zhang, C

    Y . Zhang, C. Yu, F. Fabiani, Neural network-based identification of state- space switching nonlinear systems, arXiv preprint arXiv:2503.10114 (2025). doi:10.48550/arXiv.2503.10114

    work page doi:10.48550/arxiv.2503.10114 2025

  24. [27]

    A. E. Sertba¸ s, T. Kumbasar, Stable-by-design neural network-based LPV state-space models for system identification, arXiv preprint arXiv:2510.24757 (2025). doi:10.48550/arXiv.2510.24757

    work page doi:10.48550/arxiv.2510.24757 2025

  25. [28]

    M. H. Mansur, T. Kumbasar, SOLIS: Physics-informed learning of interpretable neural surrogates for nonlinear systems, arXiv preprint arXiv:2604.14879 (2026). doi:10.48550/arXiv.2604.14879

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.2604.14879 2026

  26. [29]

    E. B. Fox, E. B. Sudderth, M. I. Jordan, A. S. Willsky, A sticky HDP- HMM with application to speaker diarization, Annals of Applied Statis- tics 5 (2011) 1020–1056. doi:10.1214/10-AOAS395

    work page doi:10.1214/10-aoas395 2011

  27. [30]

    Agarwal, L

    R. Agarwal, L. Melnick, N. Frosst, X. Zhang, B. Lengerich, R. Caruana, G. E. Hinton, Neural additive models: Interpretable machine learning with neural nets, in: Advances in Neural Information Processing Systems, volume 34, 2021

    work page 2021

  28. [31]

    Linear predictors for nonlin- ear dynamical systems: Koopman operator meets model predictive control.Automatica, 93:149–160, 2018

    M. Korda, I. Mezi ´c, Linear predictors for nonlinear dynamical systems: Koopman operator meets model predictive control, Automatica 93 (2018) 149–160. doi:10.1016/j.automatica.2018.03.046

    work page doi:10.1016/j.automatica.2018.03.046 2018

  29. [32]

    S. L. Brunton, M. Budiši´c, E. Kaiser, J. N. Kutz, Modern Koopman theory for dynamical systems, SIAM Review 64 (2022) 229–340. doi:10.1137/ 21M1401243

    work page 2022

  30. [33]

    S. L. Brunton, J. L. Proctor, J. N. Kutz, Sparse identification of nonlinear dynamics with control (SINDYc), IFAC-PapersOnLine 49 (2016) 710–

    work page 2016

  31. [34]

    doi:10.1016/j.ifacol.2016.10.249

    work page doi:10.1016/j.ifacol.2016.10.249 2016

  32. [35]

    Lusch, J

    B. Lusch, J. N. Kutz, S. L. Brunton, Deep learning for universal linear embeddings of nonlinear dynamics, Nature Communications 9 (2018)

    work page 2018

  33. [36]

    doi:10.1038/s41467-018-07210-0

    work page doi:10.1038/s41467-018-07210-0

  34. [37]

    Perez, F

    E. Perez, F. Strub, H. de Vries, V . Dumoulin, A. C. Courville, FiLM: Visual reasoning with a general conditioning layer, in: Proceedings of the AAAI Conference on Artificial Intelligence, volume 32, 2018. doi:10. 1609/aaai.v32i1.11671

    work page 2018

  35. [38]

    D. Ha, A. M. Dai, Q. V . Le, HyperNetworks, in: International Confer- ence on Learning Representations (ICLR), 2017. doi:10.48550/arXiv. 1609.09106, arXiv:1609.09106

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv 2017

  36. [39]

    A. Gu, K. Goel, C. Ré, Efficiently modeling long sequences with struc- tured state spaces, in: International Conference on Learning Representa- tions, 2022

    work page 2022

  37. [40]

    S. M. Lundberg, S.-I. Lee, A unified approach to interpreting model pre- dictions, in: Advances in Neural Information Processing Systems, vol- ume 30, 2017

    work page 2017

  38. [41]

    Lindemann, M

    L. Lindemann, M. Cleaveland, G. Shim, G. J. Pappas, Safe planning in dynamic environments using conformal prediction, IEEE Robotics and Automation Letters 8 (2023) 5116–5123. doi:10.1109/LRA.2023. 3292071

    work page doi:10.1109/lra.2023 2023

  39. [42]

    Rahaman, J

    K. Rahaman, J. V . Deshmukh, A. R. Hota, L. Lindemann, When envi- ronments shift: Safe planning with generative priors and robust confor- mal prediction, arXiv preprint arXiv:2602.12616 (2026). doi:10.48550/ arXiv.2602.12616. 16

    work page arXiv 2026

  40. [43]

    Y . Xu, W. Guo, Z. Wei, Selective conformal risk control, arXiv preprint arXiv:2512.12844 (2025). doi:10.48550/arXiv.2512.12844

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.2512.12844 2025

  41. [44]

    T. Yu, G. Thomas, L. Yu, S. Ermon, J. Zou, S. Levine, C. Finn, T. Ma, MOPO: Model-based offline policy optimization, in: Advances in Neu- ral Information Processing Systems 33 (NeurIPS 2020), 2020. doi:10. 48550/arXiv.2005.13239

    work page arXiv 2020

  42. [45]

    Y . Zhao, B. Hoxha, G. Fainekos, J. V . Deshmukh, L. Lindemann, Ro- bust conformal prediction for STL runtime verification under distribution shift, in: Proceedings of the ACM/IEEE 15th International Conference on Cyber-Physical Systems (ICCPS), 2024, pp. 169–179. doi:10.1109/ ICCPS61052.2024.00022

    work page arXiv 2024

  43. [46]

    V . Lin, R. Kaur, Y . Yang, S. Dutta, Y . Kantaros, A. Roy, S. Jha, O. Sokol- sky, I. Lee, Safety monitoring for learning-enabled cyber-physical sys- tems in out-of-distribution scenarios, arXiv preprint arXiv:2504.13478 (2025). doi:10.48550/arXiv.2504.13478

    work page doi:10.48550/arxiv.2504.13478 2025

  44. [47]

    L. Kötz, J. Sjöberg, K. Åkesson, Optimal control-based falsification of learnt dynamics via neural ODEs and symbolic regression, arXiv preprint arXiv:2602.00031 (2026). doi:10.48550/arXiv.2602.00031

    work page doi:10.48550/arxiv.2602.00031 2026

  45. [48]

    H. Yin, Y . Chen, J. Zhou, Y . Xie, Q. Wei, Z. Xu, A probabilistic deep learning approach to enhance the prediction of wastewater treatment plant effluent quality under shocking load events, Water Research X (2025). doi:10.1016/j.wroa.2024.100291

    work page doi:10.1016/j.wroa.2024.100291 2025

  46. [49]

    J. L. Martinez De La Hoz, M. M. Bappy, M. S. Islam, M. Marcantel, M. P. Hayes, Interpretable forecasting of dissolved oxygen leveraging a foun- dation model for proactive aeration in rural wastewater treatment systems, Water Research (2026). doi:10.1016/j.watres.2025.124931

    work page doi:10.1016/j.watres.2025.124931 2026

  47. [50]

    E. Bøhn, S. Eidnes, K. R. Jonassen, Machine learning in wastewater treatment: Insights from modelling a pilot denitrification reactor, arXiv preprint arXiv:2412.14030 (2024). doi:10.48550/arXiv.2412.14030

    work page doi:10.48550/arxiv.2412.14030 2024

  48. [51]

    Freyschmidt, S

    A. Freyschmidt, S. Köster, Novel approach for AI-based N 2O emis- sion reduction in biological wastewater treatment relying on genetic al- gorithms and neural networks, Water Science and Technology 91 (2025) 1172–1184. doi:10.2166/wst.2025.060

    work page doi:10.2166/wst.2025.060 2025

  49. [52]

    Aponte-Rengifo, M

    O. Aponte-Rengifo, M. Francisco, R. Vilanova, P. Vega, S. Revollar, Intelligent control of wastewater treatment plants based on model-free deep reinforcement learning, Processes 11 (2023) 2269. doi:10.3390/ pr11082269

    work page 2023

  50. [53]

    K. B. Newhart, R. W. Holloway, A. S. Hering, T. Y . Cath, Data-driven performance analyses of wastewater treatment plants: A review, Water Research 157 (2019) 498–513. doi:10.1016/j.watres.2019.03.030

    work page doi:10.1016/j.watres.2019.03.030 2019

  51. [54]

    Flores-Alsina, L

    X. Flores-Alsina, L. Corominas, L. Snip, P. A. Vanrolleghem, Including greenhouse gas emissions during benchmarking of wastewater treatment plant control strategies, Water Research 45 (2011) 4700–4710. doi:10. 1016/j.watres.2011.04.040

    work page 2011

  52. [55]

    G. Sin, K. V . Gernaey, M. B. Neumann, M. C. M. van Loosdrecht, W. Gu- jer, Uncertainty analysis in WWTP model applications: A critical discus- sion using an example from design, Water Research 43 (2009) 2894–

    work page 2009

  53. [56]

    doi:10.1016/j.watres.2009.03.048

    work page doi:10.1016/j.watres.2009.03.048 2009

  54. [57]

    S. J. Qin, Survey on data-driven industrial process monitoring and diag- nosis, Annual Reviews in Control 36 (2012) 220–234. doi:10.1016/j. arcontrol.2012.09.004

    work page doi:10.1016/j 2012

  55. [58]

    P. M. L. Ching, R. H. Y . So, T. Morck, Advances in soft sensors for wastewater treatment plants: A systematic review, Journal of Water Process Engineering 44 (2021) 102367. doi:10.1016/j.jwpe.2021. 102367

    work page doi:10.1016/j.jwpe.2021 2021

  56. [59]

    D. M. Cherenson, D. Panagou, Staggered integral online conformal pre- diction for safe dynamics adaptation with multi-step coverage guarantees, arXiv preprint arXiv:2604.06058 (2026). doi:10.48550/arXiv.2604. 06058

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.2604 2026

  57. [60]

    J. T. H. Smith, A. Warrington, S. W. Linderman, Simplified state space layers for sequence modeling, in: International Conference on Learning Representations (ICLR), 2023. ArXiv:2208.04933

    work page internal anchor Pith review Pith/arXiv arXiv 2023

  58. [61]

    A. Gu, T. Dao, Mamba: Linear-time sequence modeling with selective state spaces, arXiv preprint arXiv:2312.00752 (2023). doi:10.48550/ arXiv.2312.00752. 17

    work page internal anchor Pith review Pith/arXiv arXiv 2023