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Learning dynamical systems with biochemically informed neural ordinary differential equations

T0 review · 0 major / 1 minor · reviewed 2026-06-30 · grok-4.3

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Pith's one-line read Biochemically informed neural ODEs recover trajectories and process structure by mapping neural network outputs through a known stoichiometric matrix.

desk verdict BINODEs embed neural nets for unknown rates inside a fixed stoichiometric structure, which is a clean hybrid for biochemical ODEs when the network is known. read the letter →

arxiv 2605.24170 v1 pith:O56VQMG3 submitted 2026-05-22 math.DS cs.LGq-bio.QM

Luis L. Fonseca , Reinhard C. Laubenbacher , Lucas Böttcher This is my paper · ORCID
classification math.DScs.LGq-bio.QM
keywords BINODEsneuralordinarydifferentialequationsstoichiometricstructurebiochemicalmodelingdynamicalsystemsMonodmodelLotka-Volterrapharmacokineticmodels
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper proposes BINODEs that retain the stoichiometric structure of biochemical models while using neural networks to represent individual processes. This allows incorporating biological side information such as process inputs, sign constraints, and monotonicity directly into the model architecture. A sympathetic reader would care because it offers a way to model systems where the overall interaction structure is known but the exact rate laws are not, bridging mechanistic and data-driven approaches. The framework is shown to work on several standard models including Monod, Lotka-Volterra, pharmacokinetic, and ultradian endocrine systems.

What carries the argument

Neural network processes (NNPs) whose outputs are mapped through a stoichiometric matrix to obtain the state derivatives, with optional constraints for sign and monotonicity.

What would settle it

Applying the method to a biochemical system with known true rates and stoichiometry but finding that the learned neural processes do not match the true rates or that trajectory predictions fail to improve over unconstrained models.

Watch

Extended reading notes

Core claim

By representing each process with a neural network and mapping its outputs to state derivatives via a linear layer analogous to a stoichiometric matrix, BINODEs recover both the observed trajectories and the underlying process-level structure in biochemical dynamical systems while permitting the inclusion of biological constraints.

Load-bearing premise

The stoichiometric structure of the system is known in advance and neural networks can be constrained to respect biological assumptions without losing their ability to approximate the true process rates.

Editorial extensions

If this is right

  • Standard biochemical rate laws can be approximated by the constrained neural networks.
  • The model identifies individual process functions from trajectory data.
  • Known stoichiometric structure is preserved while allowing flexibility in rate forms.
  • Side information like monotonicity can be enforced without loss of approximation power in the tested cases.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Extending to systems where only part of the stoichiometry is known could allow partial mechanistic models.
  • Similar architectures might apply to other networked dynamical systems with known interaction graphs.
  • Using the learned processes for prediction in new conditions or for intervention design would be a natural next step.
  • The approach may improve upon pure black-box neural ODEs in interpretability for biological applications.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

0 major / 1 minor

Summary. The paper introduces biochemically informed neural ordinary differential equations (BINODEs), a neural-ODE architecture that preserves a known stoichiometric matrix while representing individual biochemical processes via neural network processes (NNPs). Biological side information such as process-specific inputs, sign constraints, and monotonicity assumptions can be incorporated directly. The work characterizes approximation properties of NNPs for standard biochemical rate laws and reports recovery of both trajectories and process-level structure on Monod, Lotka–Volterra, pharmacokinetic, and ultradian endocrine models.

Significance. If the empirical results and approximation characterizations hold, BINODEs supply a principled hybrid between fully mechanistic and black-box dynamical models for biochemical systems. The explicit retention of stoichiometry together with the stated characterization of NNP approximation power for relevant rate laws constitute concrete strengths that could support interpretability and generalization in systems biology applications.

minor comments (1)
  1. The term 'process-level structure' is used in the abstract and introduction; a concise definition or pointer to the precise metric used to assess recovery of this structure would improve clarity.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary, significance assessment, and recommendation to accept the manuscript. No major comments were raised in the report.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity

full rationale

The BINODE architecture is defined directly as a composition of a known stoichiometric matrix with independently parameterized neural network processes (NNPs) subject to explicit sign/monotonicity constraints; the approximation properties are characterized for standard rate laws and the recovery of trajectories and structure is demonstrated on concrete models. No step reduces a claimed prediction or uniqueness result to a fitted parameter or self-citation by construction, and the central claims rest on the explicit architectural choices rather than on any re-derivation of inputs.

Assumptions & free parameters 0 free parameters · 1 assumptions · 1 invented entities

The model introduces NNPs as a way to handle unknown rates while assuming known stoichiometry and standard neural network approximation capabilities.

assumptions (1)
  • domain assumption The stoichiometric structure of the biochemical system is known and can be represented by a matrix.
    The architecture relies on this to map NN outputs to state derivatives.
invented entities (1)
  • Neural network processes (NNPs)
    purpose: To represent unknown functional forms of biochemical processes.
    New component introduced in the model architecture.

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Cite this review

Pith. "Pith review of Learning dynamical systems with biochemically informed neural ordinary differential equations." pith.science (2026). https://pith.science/paper/O56VQMG3

@misc{pith2026260524170,
  author       = {Pith},
  title        = {Pith review of: Learning dynamical systems with biochemically informed neural ordinary differential equations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/O56VQMG3}},
  note         = {Machine review of arXiv:2605.24170}
}
read the original abstract

Ordinary differential equation models of biochemical reactions are often formulated as stoichiometric systems in which the dynamics arise from a collection of interacting processes. A central challenge is that the functional form of each process is rarely known a priori and may be difficult to infer from data. We propose biochemically informed neural ordinary differential equations (BINODEs), a neural-ODE framework that retains the stoichiometric structure of mechanistic models while representing individual processes by neural networks. In BINODEs, the outputs of neural network processes (NNPs) are mapped to state derivatives through a linear layer analogous to a stoichiometric matrix. This architecture allows biological side information, such as process-specific inputs, sign constraints, and monotonicity assumptions, to be built directly into the model. We characterize the approximation properties of NNPs for several standard biochemical rate laws and show that the proposed framework recovers both trajectories and process-level structure in Monod, Lotka--Volterra, pharmacokinetic, and ultradian endocrine models. These results suggest that BINODEs offer a useful compromise between mechanistic interpretability and data-driven flexibility for modeling partially known biochemical or biological dynamical systems.

Figures

Figures reproduced from arXiv: 2605.24170 by the authors.

Figure 1
Figure 1. FIG. 1. Schematic representation of a neural network process (NNP), implemented as a feedforward network with input [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Approximation of three 1D target processes by [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Approximation of three 2D target processes by [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figures from the paper (10 more)
Figure 4
Figure 4. Figure 4: FIG. 4. Schematics of BINODEs. (a) General BINODE architecture with state variables [PITH_FULL_IMAGE:figures/full_fig_p007_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5. Schematic of the BINODE used to learn the dynamics of the Monod model. [PITH_FULL_IMAGE:figures/full_fig_p009_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6. Comparison of BINODE predictions with the reference Monod model. (a–c) Time evolution of the state variables [PITH_FULL_IMAGE:figures/full_fig_p009_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7. Schematic of the BINODE used to learn the dynamics of the Lotka–Volterra system. [PITH_FULL_IMAGE:figures/full_fig_p010_7.png]
Figure 8
Figure 8. Figure 8: FIG. 8. Comparison of BINODE predictions with the reference Lotka–Volterra model. (a–c) Time evolution of the state variables [PITH_FULL_IMAGE:figures/full_fig_p011_8.png]
Figure 9
Figure 9. Figure 9: FIG. 9. Comparison of BINODE predictions with the reference pharmacokinetics model. (a) Time evolution of the state variables [PITH_FULL_IMAGE:figures/full_fig_p011_9.png]
Figure 10
Figure 10. Figure 10: FIG. 10. Comparison of BINODE predictions with the reference ultradian endocrine model. (a) Time evolution of the state [PITH_FULL_IMAGE:figures/full_fig_p013_10.png]
Figure 11
Figure 11. Figure 11: FIG. 11. Mean training time for neural network models with varying architectures used to approximate three 1D target [PITH_FULL_IMAGE:figures/full_fig_p017_11.png]
Figure 12
Figure 12. Figure 12: FIG. 12. Mean training time for neural network models with varying architectures used to approximate three 2D target [PITH_FULL_IMAGE:figures/full_fig_p017_12.png]
Figure 13
Figure 13. Figure 13: FIG. 13. Application of the BINODE to empirical biodegradation data. (a) Time evolution of the state variables [PITH_FULL_IMAGE:figures/full_fig_p019_13.png]

Discussion (0). Continue with ORCID to comment.

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Works this paper leans on

69 extracted references · 69 canonical work pages

  1. [1]

    Die Kinetik der Invertinwirkung,

    L. Michaelis and M. L. Menten, “Die Kinetik der Invertinwirkung,” Biochemische Zeitschrift, vol. 49, no. 333-369, p. 352, 1913

    work page 1913

  2. [2]

    Henri, Lois g´ en´ erales de l’action des diastases

    V. Henri, Lois g´ en´ erales de l’action des diastases. Librairie Scientifique A. Hermann, 1903

    work page 1903

  3. [3]

    Th´ eorie g´ en´ erale de l’action de quelques diastases par Victor Henri [CR Acad. Sci. Paris 135 (1902) 916-919],

    V. Henri, “Th´ eorie g´ en´ erale de l’action de quelques diastases par Victor Henri [CR Acad. Sci. Paris 135 (1902) 916-919],” Comptes Rendus Biologies, vol. 329, no. 1, pp. 47–50, 2006

    work page 1902

  4. [4]

    One hundred years of michaelis–menten kinetics,

    A. Cornish-Bowden, “One hundred years of michaelis–menten kinetics,” Perspectives in Science, vol. 4, pp. 3–9, 2015

    work page 2015

  5. [5]

    Studier over affiniteten,

    P. Waage and C. Guldberg, “Studier over affiniteten,” Forhandlinger i Videnskabs-selskabet i Christiania , vol. 1, pp. 35–45, 1864

    work page

  6. [6]

    150 years of the mass action law,

    E. O. Voit, H. A. Martens, and S. W. Omholt, “150 years of the mass action law,” PLOS Computational Biology , vol. 11, no. 1, p. e1004012, 2015

    work page 2015

  7. [7]

    A. J. Lotka, Elements of physical biology . Williams & Wilkins, 1925

    work page 1925

  8. [8]

    Volterra, Variazioni e fluttuazioni del numero d’individui in specie animali conviventi , vol

    V. Volterra, Variazioni e fluttuazioni del numero d’individui in specie animali conviventi , vol. 2. Societ´ a anonima tipografica” Leonardo da Vinci”, 1927

    work page 1927

Show all 69 references
  1. [9]

    A contribution to the mathematical theory of epidemics,

    W. O. Kermack and A. G. McKendrick, “A contribution to the mathematical theory of epidemics,” Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character , vol. 115, no. 772, pp. 700–721, 1927

    1927

  2. [10]

    Uniformly accurate nonlinear transmission rate models arising from disease spread through pair contacts,

    J. Wylie and T. Chou, “Uniformly accurate nonlinear transmission rate models arising from disease spread through pair contacts,” Physical Review E, vol. 103, no. 3, p. 032306, 2021

    2021

  3. [11]

    Biochemical systems analysis: I. Some mathematical properties of the rate law for the component enzymatic reactions,

    M. A. Savageau, “Biochemical systems analysis: I. Some mathematical properties of the rate law for the component enzymatic reactions,” Journal of Theoretical Biology, vol. 25, no. 3, pp. 365–369, 1969

    1969

  4. [12]

    Biochemical systems analysis: III. Dynamic solutions using a power-law approximation,

    M. A. Savageau, “Biochemical systems analysis: III. Dynamic solutions using a power-law approximation,” Journal of Theoretical Biology, vol. 26, no. 2, pp. 215–226, 1970

    1970

  5. [13]

    Biochemical Systems Theory: A Review,

    E. O. Voit, “Biochemical Systems Theory: A Review,” ISRN Biomathematics, vol. 2013, 2013

    2013

  6. [14]

    Dynamic simulation and metabolic re-design of a branched pathway using linlog kinetics,

    D. Visser and J. J. Heijnen, “Dynamic simulation and metabolic re-design of a branched pathway using linlog kinetics,” Metabolic engineering, vol. 5, no. 3, pp. 164–176, 2003

    2003

  7. [15]

    Myc dosage compensation is mediated by mirna-transcription factor interactions in aneuploid cancer,

    M. Ac´ on, C. Geiß, J. Torres-Calvo, D. Bravo-Estupi˜ nan, G. Oviedo, J. L. Arias-Arias, L. A. Rojas-Matey, B. Edwin, G. V´ asquez-Vargas, Y. Oses-Vargas,et al., “Myc dosage compensation is mediated by mirna-transcription factor interactions in aneuploid cancer,” IScience, vol...

    2021

  8. [16]

    Partition analysis and concept of net rate constants as tools in enzyme kinetics,

    W. Cleland, “Partition analysis and concept of net rate constants as tools in enzyme kinetics,” Biochemistry, vol. 14, no. 14, pp. 3220–3224, 1975

    1975

  9. [17]

    A schematic method of deriving the rate laws for enzyme-catalyzed reactions,

    E. L. King and C. Altman, “A schematic method of deriving the rate laws for enzyme-catalyzed reactions,” The Journal of Physical Chemistry, vol. 60, no. 10, pp. 1375–1378, 1956

    1956

  10. [18]

    A note on the kinetics of enzyme action,

    G. E. Briggs and J. B. S. Haldane, “A note on the kinetics of enzyme action,” Biochemical Journal, vol. 19, no. 2, p. 338, 1925

    1925

  11. [19]

    The possible effects of the aggregation of the molecules of hemoglobin on its dissociation curves,

    A. V. Hill, “The possible effects of the aggregation of the molecules of hemoglobin on its dissociation curves,” The Journal of Physiology, vol. 40, pp. iv–vii, 1910

    1910

  12. [20]

    The Hill equation and the origin of quantitative pharmacology,

    R. Gesztelyi, J. Zsuga, A. Kemeny-Beke, B. Varga, B. Juhasz, and A. Tosaki, “The Hill equation and the origin of quantitative pharmacology,” Archive for history of exact sciences , vol. 66, pp. 427–438, 2012

    2012

  13. [21]

    The control of flux,

    H. Kacser, “The control of flux,” in Symp Soc Exp Biol , vol. 27, p. 65, 1973

    1973

  14. [22]

    A linear steady-state treatment of enzymatic chains: general properties, control and effector strength,

    R. Heinrich and T. A. Rapoport, “A linear steady-state treatment of enzymatic chains: general properties, control and effector strength,” European Journal of Biochemistry, vol. 42, no. 1, pp. 89–95, 1974

    1974

  15. [23]

    Bringing metabolic networks to life: convenience rate law and thermodynamic constraints,

    W. Liebermeister and E. Klipp, “Bringing metabolic networks to life: convenience rate law and thermodynamic constraints,” Theoretical Biology and Medical Modelling, vol. 3, pp. 1–13, 2006

    2006

  16. [24]

    Modular rate laws for enzymatic reactions: thermodynamics, elasticities and implementation,

    W. Liebermeister, J. Uhlendorf, and E. Klipp, “Modular rate laws for enzymatic reactions: thermodynamics, elasticities and implementation,” Bioinformatics, vol. 26, no. 12, pp. 1528–1534, 2010

    2010

  17. [25]

    Cooperativity and saturation in biochemical networks: a saturable formalism using Taylor series approximations,

    A. Sorribas, B. Hern´ andez-Bermejo, E. Vilaprinyo, and R. Alves, “Cooperativity and saturation in biochemical networks: a saturable formalism using Taylor series approximations,” Biotechnology and Bioengineering, vol. 97, no. 5, pp. 1259–1277, 2007

    2007

  18. [26]

    Comparison of unstructured kinetic bacterial growth models,

    M. Muloiwa, S. Nyende-Byakika, and M. Dinka, “Comparison of unstructured kinetic bacterial growth models,” South African Journal of Chemical Engineering , vol. 33, pp. 141–150, 2020

    2020

  19. [27]

    Monod, Recherches sur la croissance des cultures bact´ eriennes

    J. Monod, Recherches sur la croissance des cultures bact´ eriennes. PhD thesis, Universit´ e de Paris, 1941

    1941

  20. [28]

    La technique de culture continue, th {´ e} orie et applications,

    J. Monod, “La technique de culture continue, th {´ e} orie et applications,” in Annales de l’Institut Pasteur , vol. 79, pp. 390–410, 1950

    1950

  21. [29]

    Haldane, Enzymes

    J. Haldane, Enzymes. Cambridge, MA, USA: MIT Press, 1965

    1965

  22. [30]

    A mathematical model for the continuous culture of microorganisms utilizing inhibitory substrates,

    J. F. Andrews, “A mathematical model for the continuous culture of microorganisms utilizing inhibitory substrates,” Biotechnology and Bioengineering, vol. 10, no. 6, pp. 707–723, 1968

    1968

  23. [31]

    Moser, The dynamics of bacterial populations maintained in the chemostat

    H. Moser, The dynamics of bacterial populations maintained in the chemostat. Carnegie Institution of Washington, 1958

    1958

  24. [32]

    The components of predation as revealed by a study of small mammal predation of the European pine sawfly,

    C. S. Holling, “The components of predation as revealed by a study of small mammal predation of the European pine sawfly,” The Canadian Entomologist , vol. 91, pp. 293–320, 1959

    1959

  25. [33]

    Some characteristics of simple types of predation and parasitism,

    C. S. Holling, “Some characteristics of simple types of predation and parasitism,” The Canadian Entomologist , vol. 91, no. 7, pp. 385–398, 1959. 22

    1959

  26. [34]

    A derivation of Holling’s type I, II and III functional responses in predator–prey systems,

    J. Dawes and M. Souza, “A derivation of Holling’s type I, II and III functional responses in predator–prey systems,” Journal of Theoretical Biology, vol. 327, pp. 11–22, 2013

    2013

  27. [35]

    Biologically informed NeuralODEs for genome-wide regulatory dynamics,

    I. Hossain, V. Fanfani, J. Fischer, J. Quackenbush, and R. Burkholz, “Biologically informed NeuralODEs for genome-wide regulatory dynamics,” Genome Biology, vol. 25, no. 1, p. 127, 2024

    2024

  28. [36]

    Universal differential equations for systems biology: Current state and open problems,

    M. Philipps, N. Schmid, and J. Hasenauer, “Universal differential equations for systems biology: Current state and open problems,” bioRxiv, pp. 2024–11, 2024

    2024

  29. [37]

    Physiology-informed regularisation enables training of universal differential equation systems for biological applications,

    M. de Rooij, B. Erd˝ os, N. A. van Riel, and S. D. O’Donovan, “Physiology-informed regularisation enables training of universal differential equation systems for biological applications,” PLOS Computational Biology , vol. 21, no. 1, p. e1012198, 2025

    2025

  30. [38]

    A hybrid neural ordinary differential equation model of the cardiovascular system,

    G. Grigorian, S. V. George, S. Lishak, R. J. Shipley, and S. Arridge, “A hybrid neural ordinary differential equation model of the cardiovascular system,” Journal of the Royal Society Interface , vol. 21, no. 212, p. 20230710, 2024

    2024

  31. [39]

    Modeling chemical reaction networks using neural ordinary differential equations,

    A. C. Th¨ oni, W. E. Robinson, Y. Bachrach, W. T. Huck, and T. Kachman, “Modeling chemical reaction networks using neural ordinary differential equations,” Journal of Chemical Information and Modeling , 2025

    2025

  32. [40]

    Control of dynamical systems with neural networks,

    L. B¨ ottcher, “Control of dynamical systems with neural networks,”Nonlinear Dynamics, vol. 114, no. 2, p. 79, 2026

    2026

  33. [41]

    Learning dynamical systems with side information,

    A. A. Ahmadi and B. E. Khadir, “Learning dynamical systems with side information,” in Proceedings of the 2nd Annual Conference on Learning for Dynamics and Control, L4DC 2020, Online Event, Berkeley, CA, USA, 11-12 June 2020 (A. M. Bayen, A. Jadbabaie, G. J. Pappas, P. A. Parr...

    2020

  34. [42]

    Learning dynamical systems with side information,

    A. A. Ahmadi and B. E. Khadir, “Learning dynamical systems with side information,” SIAM Review, vol. 65, no. 1, pp. 183–223, 2023

    2023

  35. [43]

    Optimal control of agent-based models via surrogate modeling,

    L. L. Fonseca, L. B¨ ottcher, B. Mehrad, and R. C. Laubenbacher, “Optimal control of agent-based models via surrogate modeling,” PLOS Computational Biology , vol. 21, no. 1, p. e1012138, 2025

    2025

  36. [44]

    Interpretable polynomial neural ordinary differential equations,

    C. Fronk and L. Petzold, “Interpretable polynomial neural ordinary differential equations,” Chaos: An Interdisciplinary Journal of Nonlinear Science , vol. 33, no. 4, 2023

    2023

  37. [45]

    Control of medical digital twins with artificial neural networks,

    L. B¨ ottcher, L. L. Fonseca, and R. C. Laubenbacher, “Control of medical digital twins with artificial neural networks,” Philosophical Transactions A, vol. 383, no. 2292, p. 20240228, 2025

    2025

  38. [46]

    Learning effective stochastic differential equations from microscopic simulations: Linking stochastic numerics to deep learning,

    F. Dietrich, A. Makeev, G. Kevrekidis, N. Evangelou, T. Bertalan, S. Reich, and I. G. Kevrekidis, “Learning effective stochastic differential equations from microscopic simulations: Linking stochastic numerics to deep learning,” Chaos: An Interdisciplinary Journal of Nonlinear...

    2023

  39. [47]

    Reconstructing noisy gene regulation dynamics using extrinsic-noise-driven neural stochastic differential equations,

    J. Zhang, X. Li, X. Guo, Z. You, L. B¨ ottcher, A. Mogilner, A. Hoffmann, T. Chou, and M. Xia, “Reconstructing noisy gene regulation dynamics using extrinsic-noise-driven neural stochastic differential equations,” PLOS Computational Biology , vol. 21, no. 9, p. e1013462, 2025

    2025

  40. [48]

    AI-Aristotle: A physics-informed framework for systems biology gray-box identification,

    N. Ahmadi Daryakenari, M. De Florio, K. Shukla, and G. E. Karniadakis, “AI-Aristotle: A physics-informed framework for systems biology gray-box identification,” PLOS Computational Biology , vol. 20, pp. 1–33, 03 2024

    2024

  41. [49]

    Why RELU units sometimes die: Analysis of single-unit error backpropagation in neural networks,

    S. C. Douglas and J. Yu, “Why RELU units sometimes die: Analysis of single-unit error backpropagation in neural networks,” in 52nd Asilomar Conference on Signals, Systems, and Computers, ACSSC 2018, Pacific Grove, CA, USA, October 28-31, 2018 (M. B. Matthews, ed.), pp. 864–868...

    2018

  42. [50]

    Multilayer feedforward networks are universal approximators,

    K. Hornik, M. Stinchcombe, and H. White, “Multilayer feedforward networks are universal approximators,” Neural networks, vol. 2, no. 5, pp. 359–366, 1989

    1989

  43. [51]

    The expressive power of neural networks: A view from the width,

    Z. Lu, H. Pu, F. Wang, Z. Hu, and L. Wang, “The expressive power of neural networks: A view from the width,” in Advances in Neural Information Processing Systems 30: Annual Conference on Neural Information Processing Systems 2017, December 4-9, 2017, Long Beach, CA, USA (I. Gu...

    2017

  44. [52]

    Benefits of depth in neural networks,

    M. Telgarsky, “Benefits of depth in neural networks,” in Proceedings of the 29th Conference on Learning Theory, COLT 2016, New York, USA, June 23-26, 2016 (V. Feldman, A. Rakhlin, and O. Shamir, eds.), vol. 49 of JMLR Workshop and Conference Proceedings, pp. 1517–1539, JMLR.org, 2016

    2016

  45. [53]

    Biochemical systems analysis: II. The steady-state solutions for an n-pool system using a power-law approximation,

    M. A. Savageau, “Biochemical systems analysis: II. The steady-state solutions for an n-pool system using a power-law approximation,” Journal of Theoretical Biology, vol. 25, no. 3, pp. 370–379, 1969

    1969

  46. [54]

    Input convex neural networks,

    B. Amos, L. Xu, and J. Z. Kolter, “Input convex neural networks,” in Proceedings of the 34th International Conference on Machine Learning, ICML 2017, Sydney, NSW, Australia, 6-11 August 2017 (D. Precup and Y. W. Teh, eds.), vol. 70 of Proceedings of Machine Learning Research, ...

    2017

  47. [55]

    PySINDy: A Python package for the sparse identification of nonlinear dynamical systems from data,

    B. de Silva, K. Champion, M. Quade, J.-C. Loiseau, J. Kutz, and S. Brunton, “PySINDy: A Python package for the sparse identification of nonlinear dynamical systems from data,” Journal of Open Source Software , vol. 5, no. 49, p. 2104, 2020

    2020

  48. [56]

    PySINDy: A comprehensive python package for robust sparse system identification,

    A. A. Kaptanoglu, B. M. de Silva, U. Fasel, K. Kaheman, A. J. Goldschmidt, J. Callaham, C. B. Delahunt, Z. G. Nicolaou, K. Champion, J.-C. Loiseau, J. N. Kutz, and S. L. Brunton, “PySINDy: A comprehensive python package for robust sparse system identification,” Journal of Open...

    2022

  49. [57]

    Discovering governing equations from data by sparse identification of nonlinear dynamical systems,

    S. L. Brunton, J. L. Proctor, and J. N. Kutz, “Discovering governing equations from data by sparse identification of nonlinear dynamical systems,” Proceedings of the National Academy of Sciences , vol. 113, no. 15, pp. 3932–3937, 2016

    2016

  50. [58]

    PySINDy

    A. Kaptanoglu, J. Stevens-Haas, K. Champion, B. de Silva, and M. Quade, “PySINDy.”

  51. [59]

    Extreme theory of functional connections: A fast physics-informed neural network method for solving ordinary and partial differential equations,

    E. Schiassi, R. Furfaro, C. Leake, M. De Florio, H. Johnston, and D. Mortari, “Extreme theory of functional connections: A fast physics-informed neural network method for solving ordinary and partial differential equations,” Neurocomputing, vol. 457, pp. 334–356, 2021

    2021

  52. [60]

    Physics-informed neural networks and functional interpolation for stiff chemical kinetics,

    M. De Florio, E. Schiassi, and R. Furfaro, “Physics-informed neural networks and functional interpolation for stiff chemical kinetics,” Chaos: An Interdisciplinary Journal of Nonlinear Science , vol. 32, no. 6, 2022. 23

    2022

  53. [61]

    Symbolic regression is NP-hard,

    M. Virgolin and S. P. Pissis, “Symbolic regression is NP-hard,” arXiv preprint arXiv:2207.01018 , 2022

    2022

  54. [62]

    gplearn: Genetic programming in python with a scikit-learn inspired and compatible api,

    T. Stephens, “gplearn: Genetic programming in python with a scikit-learn inspired and compatible api,” 2026. Version 0.4.3

    2026

  55. [63]

    Barnes and G

    B. Barnes and G. R. Fulford, Mathematical modelling with case studies: a differential equations approach using Maple and MATLAB. Chapman and Hall/CRC, 2011

    2011

  56. [64]

    Computer model for mechanisms underlying ultradian oscillations of insulin and glucose,

    J. Sturis, K. S. Polonsky, E. Mosekilde, and E. Van Cauter, “Computer model for mechanisms underlying ultradian oscillations of insulin and glucose,” American Journal of Physiology-Endocrinology and Metabolism , vol. 260, no. 5, pp. E801–E809, 1991

    1991

  57. [65]

    Complex coordination of multi-scale cellular responses to environmental stress,

    L. L. Fonseca, C. Sanchez, H. Santos, and E. O. Voit, “Complex coordination of multi-scale cellular responses to environmental stress,” Molecular BioSystems, vol. 7, no. 3, pp. 731–741, 2011

    2011

  58. [66]

    The origins of enzyme kinetics,

    A. Cornish-Bowden, “The origins of enzyme kinetics,” FEBS letters, vol. 587, no. 17, pp. 2725–2730, 2013

    2013

  59. [67]

    Enzymes longmans,

    J. Haldane, “Enzymes longmans,” Green and Co, UK , vol. 7, 1930

    1930

  60. [68]

    The reversible Hill equation: how to incorporate cooperative enzymes into metabolic models,

    J.-H. S. Hofmeyr and H. Cornish-Bowden, “The reversible Hill equation: how to incorporate cooperative enzymes into metabolic models,” Bioinformatics, vol. 13, no. 4, pp. 377–385, 1997

    1997

  61. [69]

    Biodegradation kinetics of benzene, toluene, and phenol as single and mixed substrates for Pseudomonas putida F1,

    K. F. Reardon, D. C. Mosteller, and J. D. Bull Rogers, “Biodegradation kinetics of benzene, toluene, and phenol as single and mixed substrates for Pseudomonas putida F1,” Biotechnology and Bioengineering, vol. 69, no. 4, pp. 385–400, 2000

    2000

Pith tools

Reviewed June 30, 2026 · model on record in the stance chip above.