PaperlayerView exact revision on arXiv

Source-linked AI summary

Equation free data-driven modelling of chaotic processes

Aurora Poggi, Marco Martens, Hjalmar Brismar, Ozan Oktem, Liviana Palmisano

arXiv:2609.08530v1math.DScs.ITmath.IT

TL;DR

When governing equations are unknown and observations are finite and noisy, the paper builds data-driven models without assuming those equations. It applies hyperbolicity criteria to identify when its scale-based Markov models provide a statistical description, and concludes that the Cell Process is chaotic with entropy approximately 2.3.

  • Problem

    For complex physical and biological processes, governing equations may be unknown or inaccessible, leaving only finite, noisy time-series data.

    Source: §1 · p. 1

  • Method

    The method constructs a discrete model from observed time-series orbits, studies recurrent dynamics across geometric scales, and uses hyperbolicity criteria to identify a suitable Markov model.

    Source: §1 · p. 1, §1.3.4 · p. 6, §2.5 · p. 14

  • Results

    Entropy approximately 2.3: the Cell Process is concluded to be chaotic and describable by a Markov chain.

    Source: §3.2 · p. 25

  • Takeaways & Limitations

    The Markov-process description allows statistical tools to be used to study the Cell Process.

    Source: §3.2 · p. 25

    Reliable statistical conclusions require sufficiently long time series, and failed validation criteria may indicate that the available data are insufficient.

    Source: The attractor · p. 6

Abstract

from arXiv · show

We introduce a method for constructing predictive models of non cyclic physical processes directly from time-series data, without assuming an underlying differential equation. The observations define a discrete evolution rule whose recurrent behaviour captures the essential dynamics of the process. Analysing this behaviour across multiple geometric scales leads to probabilistic models in the form of Markov chains. Hyperbolicity criteria identify when these models provide a consistent statistical description of the data. The method is inspired by, and illustrated through, the analysis of a biological imaging data set referred to as the Cell Process.

1 Introduction

The introduction frames equation-free modeling of chaotic, non-cyclic processes as a data-driven approach, organizing noisy observations into scale-dependent Markov models and using hyperbolicity criteria to assess their relevance. It motivates this approach for calcium oscillations and cell-signalling microscopy data, while situating it among alternatives for recovering dynamics from data and reviewing their assumptions and limitations.

1 Introduction

  • The framework constructs models directly from observations, without governing-equation assumptions, using orbits formed by concatenating observed time-series segments.The paper demonstrates the approach through cell-signalling dynamics modeled from microscopy data.Source: §1 · p. 1, §1.3 · p. 3, §1.4 · p. 7

1.1 Challenges with scientific discovery for complex systems

  • Chaotic dynamics make reliable prediction from a single fitted differential equation difficult: sensitivity to initial conditions and parameters renders both forward and inverse problems ill-posed.Small perturbations can produce markedly different trajectories, while small parameter changes can alter the observed behavior.Source: §1.1 · p. 2

1.3.1 Basic dynamical systems theory

  • Renormalization represents attractor dynamics across geometric scales as symbolic paths through partitions, producing hyperbolic models that can capture the original attractor dynamics when it is hyperbolic.The symbolic description records transitions among regions, while a physical measure connects typical trajectories to statistical predictions.Source: §1.3.1 · pp. 3–4

1.3.2 Guiding principle for constructing a generative model

  • Hyperbolicity guides model construction without requiring proof that the physical process is hyperbolic, while validation criteria select observed cycles that share statistical properties and can approximate its physical measure.The criteria also indicate whether to continue construction or whether data insufficiency or a more fundamental issue may be present.Source: §1.3.2 · p. 5

The attractor:

  • Scale-dependent Markov chains arise from periodic orbits treated as cycles at a geometric scale, and entropy variation helps identify whether a scale is too coarse or too fine.The resulting chain uses measured states for probabilistic predictions; reliable statistical conclusions require sufficiently long time series, and failed validation may indicate insufficient data.Source: The attractor · pp. 5–6

1.3.4 Detecting and recovering emergent properties from data

  • Hyperbolicity criteria test whether observed dynamics behave like a Markov chain and identify a suitable scale; constructing chains alone is not evidence of such behavior.A random time series would almost certainly fail the criteria; when they are met, the method treats the Markov behavior as an emergent property and recovers the chain stably.Source: §1.3.4 · p. 6

1.4 Application to scientific discovery in cell signalling dynamics

  • The method is proposed for modeling calcium-level oscillations in living cell populations from live-imaging data, without introducing a priori differential equations.The authors note that in situ fluorescence microscopy can track dynamically changing calcium levels and point to prior differential-equation modeling of such oscillations.Source: §1.4 · p. 7

1.5 Related methods

  • Dynamical-system recovery divides into parametric methods, which fix a vector-field family, and equation-free methods, which return surrogates such as Markov approximations or invariant sets.The paper places its method in the Markov-approximation category.Source: §1.5 · p. 7

1.5.1 Parametric methods

  • Classical parametric recovery can be ill-posed, while sparse regression broadens the model class but depends on a dictionary that sparsifies the dynamics and a rule for choosing regularization.SINDy seeks a sparse combination of dictionary functions; its ℓ0 formulation is non-convex and NP-hard, and equivalence to convex relaxation requires difficult-to-verify coherence conditions.Source: §1.5.1 · p. 8
  • Symbolic regression searches expression trees beyond fixed dictionaries, whereas neural parametrizations sacrifice interpretability and can still face extrapolation and hallucination barriers.Symbolic-regression search is NP-hard; SINDy is a more restricted linear ansatz over a fixed library, trading expressiveness for convexity and speed.Source: §1.5.1 · p. 8

1.5.2 Equation-free methods

  • Equation-free methods study identifiable features and state reconstruction, but delay embeddings may be ill-conditioned and Koopman approximations face challenges from infinite-dimensionality and finite compression.Identifiability results cover linear systems and broad nonlinear ODE/PDE classes, while a cited result identifies chaos as a key ingredient; EDMD is reported to converge strongly as data and dictionary sizes grow.Source: §1.5.2 · p. 9
  • Beyond the Lyapunov horizon, distributional training targets the invariant measure; the paper likewise builds a Markov model from that measure, while Conley methods certify qualitative dynamics on isolating neighborhoods.Conley-based conclusions use outer approximations on cubical grids, and most reviewed equation-free methods rely on numerical dynamical-systems algorithms.Source: §1.5.2 · p. 9

2 Detailed description of the method

The method organizes observed dynamics into Markov chains across geometric scales, accepting them when validation criteria are met within an application-defined uncertainty threshold. It builds a discrete evolution from time-series data, isolates recurrent behaviour, and analyzes the attractor at progressively finer scales.

2 Detailed description of the method

  • At consecutive geometric scales, the method constructs Markov chains and validates them against an uncertainty threshold set by application needs.The model is one of the chains whose validation criteria are satisfied.Source: §2 · pp. 9–10

2.1 Time-series

  • When data dimension is too large relative to series length, continuous-valued observations can be rearranged into lower-dimensional projections, trading state-space resolution for more time-series samples.This reduction is optional when the original dimension is manageable.Source: §2.2 · p. 10

2.2 Projected time-series

  • Projected time series increase the number of series while reducing resolution insignificantly, and allow chain recurrence to be incorporated so Markov chains can model the data.The projections should be physically relevant; the paper notes there is no general rule for choosing informative ones.Source: §2.2 · p. 10

2.3 The discrete differential equation

  • The projected observations form a state space X of size K × T, with a map f assigning each point its successor.Within a series, the next observation is the natural successor; at its end, the successor is chosen as a closest point in a future portion of the data.Source: §2.3 · p. 11

2.4 The attractor

  • The finite system’s attractor is the invariant set of periodic points, where the essential aspects of its dynamics occur.For the Cell Process, the attractor’s projection has clearly defined features unlike the projection of the full state space.Source: §2.4 · p. 13, Figure 6 · p. 13
  • The attractor contains 69,846 points, less than 2% of the 4,318,000-point state space, and consists of 30 cycles.The remaining 98% lies in the basin and is not relevant to recurrence.Source: §2.4 · p. 13

2.5 Renormalization

  • Renormalization studies attractor geometry across scales through nested domains and first-return maps, with each domain defining dynamics at its scale.The Cell Process has return-time distribution values of 0 or 1, so each point in each domain returns with its own time.Source: §2.5 · pp. 14–15

2.6 Dynamical partitions and renormalization sub-shifts

The section builds scale-dependent symbolic descriptions from renormalization partitions, then incorporates measurement uncertainty and tests whether observed cycles have the properties expected of physically relevant dynamics. It uses these cycles to motivate a statistical description through a physical measure.

2.6 Dynamical partitions and renormalization sub-shifts

  • Each renormalization domain defines a geometric-scale partition, and its return-time distribution determines a rooted graph and associated finite-type sub-shift.Successive partitions refine one another, giving a combinatorial description of the attractor across scales.Source: §2.6 · pp. 15–16

2.7 Revisiting the attractor: chain recurrence

  • Measurement uncertainty makes successors ambiguous, so chain recurrence includes all orbits whose next state lies in the uncertainty neighborhood of the discrete map's successor.The physical orbits are a subset of these possible orbits; the construction restricts ambivalence to renormalization boxes.Source: §2.7 · pp. 16–17
  • Validation Criterion I requires every observed cycle to intersect each renormalization box, matching the shared crossing property expected of cycles in a hyperbolic system.The authors interpret this criterion as indicating that the physical process uses deep cycles.Source: §2.7 · p. 17

2.8 The physical measure

  • A physical measure captures the frequency with which a typical orbit visits open sets, and constructing this measure is the stated goal for making meaningful predictions.The section relates independence from the typical starting point to ergodicity.Source: §2.8 · pp. 17–18

2.9 Cycle measures

  • Validation Criterion II tests whether observed cycles pass through renormalization boxes with approximately equal frequency, indicating statistical organization beyond merely crossing every box.For the Cell signalling example, the reported value is δII = 0.1040.Source: §2.9 · p. 19

2.10 A Markov chain as model

The section turns scale-dependent symbolic systems into probabilistic Markov chains by assigning measures derived from observed cycles. It uses shared entropy profiles and validation criteria to identify a scale at which a chain can serve as the process model.

2.10 A Markov chain as model

  • At each scale, the dynamical partition supplies a finite-type sub-shift, which is enriched with a probability measure built from observed cycles and their cycle measures.This yields the probabilistic Markov chain used to model the physical process.Source: §2.10 · p. 19
  • The transition rules assign zero probability to absent graph edges and probability one to outgoing edges from non-root vertices, while root transitions encode the remaining probabilities.Together with the invariant measure, these rules define the transition matrix and probability vector.Source: §2.10 · p. 19
  • The transition matrix has probability-vector rows, and its stationary vector is a left eigenvector; its powers converge exponentially to a matrix whose rows equal that vector.These properties are stated for the constructed Markov chain.Source: §2.10 · p. 20
  • Validation Criterion III identifies a plateau where observed cycles share essentially the same entropy, supporting use of the corresponding measure as the model measure.For the Cell signalling example, the reported plateau is S = [25, 36] for δIII = 0.4234.Source: §2.10 · pp. 21–22
  • A Markov chain at any scale in the entropy plateau S can serve as the process model, with its scale k* determining the model's entropy.The paper suggests choosing k* near the middle of S in practice.Source: §2.10 · p. 21

2.11 Further possible criteria for hyperbolicity

This section adds a criterion based on whether a finite collection of cycle measures agrees across a projected grid, and describes estimating observable averages from that projection. In the Cell Signalling example, individual cycle measures differ substantially, so the collection as a whole is used to capture statistical behavior.

2.11 Further possible criteria for hyperbolicity

  • Projecting onto coordinates relevant to an observable and approximating its integral over a finite grid yields a prediction with uncertainty bounded by δ|φ|C1.The grid consists of cubes of width δ in the reduced space.Source: §2.11 · pp. 22–23
  • Validation Criterion IV tests whether a finite collection of cycles assigns sufficiently similar weights to boxes in a projected grid.The authors present it as an additional criterion; satisfying it makes the selected cycles play the role of those expected in a hyperbolic setting.Source: §2.11 · pp. 23–24
  • In the Cell signalling example, both projections—to grid squares 152 and 153, and to 97 and 113—have two cycles satisfying Validation Criterion IV; δ2 IV = 0.3114 is reported for 152 and 153.Source: §2.11 · p. 24
  • For the Cell Signalling Process, cycle measures differ significantly, so the statistical behavior is captured by considering the collection together; a more accurate approximation would require more data.The text notes that available data do not support a sufficiently long cycle for accurately approximating the physical measure.Source: §2.11 · p. 24

3 Cell signalling process for a kidney cell population

The section examines whether calcium-imaging time series from a kidney-cell population reveal collective signalling dynamics, using a data-derived Markov model and delayed correlations between cell signals. The model describes the process as chaotic with entropy approximately 2.3, while correlations show similar profiles across model scales and suggest possible 20-second responses or local rhythm.

3.1 Description of the experiment and the data

  • The data are 512 × 512 time-lapse images of calcium activity, treated as coupled time series to investigate emergent properties of the cell population.The recordings track intracellular calcium activity in MDCK-cell monolayers; fluorescence intensity is converted into normalized calcium traces for each pixel.Source: §3.1 · pp. 24–25

3.2 Application of the model to the Cell Signalling Process

  • The cell-signalling process is described as chaotic by a Markov chain with entropy approximately 2.3.The authors state that the Markov process allows statistical tools to be used.Source: §3.2 · p. 25
  • Delayed-correlation profiles for representative cell pairs are essentially the same across the considered models in S, indicating consistency across model scales.The analysis uses delayed correlations between signals of two cells to ask whether, and how long, it takes them to react to one another.Source: §3.2 · pp. 25–26
  • All considered cell pairs have nonzero initial correlation; it is weaker for distant cells and stronger for neighboring cells.The strongest instantaneous organization is reported for a cell correlated with itself.Source: §3.2 · p. 26
  • For pairs with stronger initial organization, a second correlation spike occurs after 20 seconds, including for two pairs described as non-neighbors.The authors suggest this might indicate one cell responding to another; the response time is the same for the two pairs.Source: §3.2 · pp. 26–27
  • A 20-second spike in a cell's self-correlation may indicate a local rhythm, while a rhythm in the whole colony remains a question for further investigation.The self-correlation spike is reported for the cell at position 152.Source: §3.2 · p. 27

4 Robustness of the model construction

The section assesses model robustness by comparing entropy across models built from different datasets and by using hyperbolicity-motivated criteria. Across the reported cell-signalling models, entropy and cycle entropy profiles are described as essentially consistent, with outlying cycles and cell movement noted as qualifications.

4 Robustness of the model construction

  • The robustness framework treats models as essentially the same when their entropies differ by no more than a prescribed threshold.The comparison is motivated by the classification theory of Markov chains; a sufficiently small validation threshold is used to assess robustness across data sets.Source: §4 · p. 27

4.1 Different data sets

  • The study compares the initial and second six-hour data blocks, then varies the number of projected time series to K = 1500 and K = 5000.The second block is treated as independent from the first-block movie; the other data characteristics are kept the same for the grid-density experiments.Source: §4 · p. 28, §4.1 · p. 28, §4.2 · p. 29

4.3 Model reliability

  • Across the cell-signalling models, entropies are essentially the same, aggregate entropy profiles are very similar, and individual cycles also share similar entropy profiles.The authors describe this consistency as remarkable given the complexity of the process.Source: §4.3 · p. 29

4.2 Different number of projected time series

  • Increasing grid density to K = 5000 did not yield longer cycles or smaller bundling widths; refining the grid may be needed to tighten the bundling constants.The reported cycle periods are mostly between 2000 and 2500, except for three cycles.Source: §4.3 · p. 29

4.3 Model reliability

  • Two outlying cycles in X6−12,1000 may be responsible for that dataset's large δIII.The authors suggest considering whether to eliminate the outlying cycles.Source: §4.3 · p. 29
  • Some cells changed location between the first and second six-hour blocks, and the authors say careful studies should incorporate this motion.All models based on the first six hours show the correlation spike after 20 seconds.Source: §4.3 · p. 29

Appendix I

Appendix I collects figures and tables for the different data sets, including validation, entropy profiles, delayed correlations, and cycle-period distributions.

Appendix I

Appendix II

Appendix II provides sketches of proofs for consequences of hyperbolicity and refers readers to further details.

Appendix II

  • The appendix sketches proofs of consequences derived from hyperbolicity and points to references for further details.Source: Appendix II · p. 33

Hyperbolicity Consequence I

For a hyperbolic system, nested renormalization boxes correspond to sub-shifts that are equivalent to the system at sufficiently large scales. Finite selections of boxes can also be intersected by collections of cycles.

Hyperbolicity Consequence I

  • At sufficiently large k, the system is equivalent to the sub-shift corresponding to its nested renormalization boxes.The boxes are indexed by 0 ≤ k < m.Source: Hyperbolicity Consequence I · p. 33
  • A finite selection of renormalization boxes is intersected by a collection of cycles.The stated result covers boxes with k < m.Source: Hyperbolicity Consequence I · p. 33
  • The proof idea is to choose nested Markov boxes and use their corresponding sub-shift of finite type as an equivalent description of the system.Source: Hyperbolicity Consequence I · p. 33

Hyperbolicity Consequence II

For a hyperbolic system with a physical measure, the passage states that finite selections of renormalization boxes admit a nonempty finite collection of cycles. The proof idea invokes approximation of invariant measures by periodic-orbit measures.

Hyperbolicity Consequence II

  • For any δ > 0 and finite selection of renormalization boxes, a nonempty finite collection of cycles exists for a hyperbolic system with physical measure μ∗.The passage does not include the further property required of that collection.Source: Hyperbolicity Consequence II · p. 33
  • The proof idea uses the fact that every invariant measure of a sub-shift of finite type can be approximated by the invariant measure supported on a periodic orbit.Source: Hyperbolicity Consequence II · p. 33

Hyperbolicity Consequence III

The cycle-associated entropy function is stated to increase at low scales and decrease at high scales, with zero entropy at both endpoints. The proof discussion relates these regimes to approximation of the physical measure and to the cycle’s shrinking loop support.

Hyperbolicity Consequence III

  • The cycle entropy hγ(k) increases for k ≤ k(0) and decreases for k ≥ k(1).The theorem states this shape for cycles in a finite collection associated with a hyperbolic system having physical measure μ∗.Source: Hyperbolicity Consequence III · p. 33
  • The proof constructs finer Markov partitions via renormalization; their sub-shifts have graphs like those in the method, with countably many loops, and shrinking boxes under hyperbolicity.Source: Hyperbolicity Consequence III · p. 33
  • For each Markov partition, weights assigned by the physical measure define a Markov measure whose entropy converges to h∗.Source: Hyperbolicity Consequence III · p. 33
  • The proof compares the cycle’s Markov measure with the physical measure on the partition at k(0), whose cycle-induced entropy is approximately h_k(0).Source: Hyperbolicity Consequence III · p. 33
  • The cycle entropy is zero at k = 0 and k = m.At k = 0 the graph has one vertex and one return branch; at sufficiently large m the return box contains only one cycle point, supported on one loop.Source: Hyperbolicity Consequence III · p. 33
  • At low k, increasing k moves weight from the root into a growing number of loops while the cycle’s Markov measure remains close to the physical-measure-induced measure, so entropy increases.Source: Hyperbolicity Consequence III · p. 34
  • At intermediate scales k(0) ≤ k ≤ k(1), the entropy h_k is approximately the physical-measure entropy h∗.The passage attributes this to approximate agreement between the Markov measures induced by the cycle measure and the physical measure.Source: Hyperbolicity Consequence III · p. 34
  • At high k, the cycle has too few points to approximate the physical-measure Markov measure well, and its entropy decreases as its support contracts to fewer loops.Source: Hyperbolicity Consequence III · p. 34

Hyperbolicity Consequence IV

For a hyperbolic system with physical measure, the passage asserts existence of a finite collection of cycles for any δ > 0 and p ∈ N. The proof idea approximates measures on a finite cover using cycle measures.

Hyperbolicity Consequence IV

  • For any δ > 0 and p ∈ N, a hyperbolic system with physical measure admits a nonempty finite collection of cycles.The passage truncates before stating the further property of that collection.Source: Hyperbolicity Consequence IV · p. 34
  • The proof idea approximates an invariant measure on the algebra generated by a finite cover of Markov boxes using cycle measures.The cover is chosen fine enough, and the approximation can be arbitrarily precise.Source: Hyperbolicity Consequence IV · p. 34

Hyperbolicity robustness consequence

For a hyperbolic system with physical measure and entropy, suitable data characteristics yield a model whose entropy is close to the system’s entropy. The proof links this result to induced Markov measures and sufficiently long typical orbits.

Hyperbolicity robustness consequence

  • For every δ, suitable data characteristics yield a model whose entropy is close to the hyperbolic system’s entropy.The theorem concerns a system with physical measure µ∗ and entropy h∗.Source: Hyperbolicity robustness consequence · p. 34
  • A small Markov box and its first return map produce a dynamical partition and renormalization graph carrying an induced stationary Markov measure.The entropy of that induced measure can be made arbitrarily close to the physical measure’s entropy.Source: Hyperbolicity robustness consequence · p. 34
  • A periodic orbit can approximate the physical measure on the partition pieces, bringing their induced Markov measures and entropies close.Source: Hyperbolicity robustness consequence · p. 34
  • Long enough typical data orbits preserve this approximation, so the resulting model’s entropy remains close to the system’s entropy.Source: Hyperbolicity robustness consequence · p. 34

Related papers

Loading 2609.08530v1…