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Physical Law Ecology: mapping multi-mechanism ecologies as the zeroth step of data-driven scientific discovery

Xiongheng Bian, Xiangyu Cui, Ma Feng, Xiaoyan Shen

arXiv:2609.08536v1cs.SCastro-ph.GAphysics.comp-ph

TL;DR

The paper addresses equation discovery’s default assumption that a system has one governing law, despite mechanisms varying across conditions and locations. It proposes Physical Law Ecology to determine mechanism count and map coexisting equations, their dominance, and succession. The paper reports that this reframes discovery as mapping mechanism ecologies, while identifying limits in operator coverage and extrapolation.

  • Problem

    Equation-discovery methods generally assume one governing law, while complex systems may undergo qualitative mechanism changes that parameter adjustment alone cannot represent.

    Source: Introduction · p. 1, Discussion · p. 18

  • Method

    Physical Law Ecology uses BIC to determine K* and maps multiple topologies, their dominance across parameter space, and their evolution.

    Source: Discussion · p. 18, Bayesian formulation: dominance field… · p. 20

  • Results

    When K*>1, the framework outputs coexisting equations, where each dominates, and how they succeed one another, information classical single-equation methods cannot provide.

    Source: Discussion · p. 18

  • Takeaways & Limitations

    The paper frames determining K* as a zeroth step that shifts discovery from finding one equation to mapping a mechanism ecology.

    Source: Discussion · p. 18

    The operator vocabulary does not yet cover trigonometric or special functions, extrapolation degrades far from training conditions, and model coverage is typically 85%–95%.

    Source: Discussion · p. 19

Abstract

from arXiv · show

Every data-driven equation discovery method assumes (implicitly and without verification) that the target system obeys a single governing law ($K{=}1$). Here we show that this assumption is the primary bottleneck limiting scientific discovery in multi-mechanism systems, and introduce Physical Law Ecology, a framework that makes $K^*$ (the number of coexisting independent mechanisms) itself the first quantity to be determined from data. The framework automatically mines a pool of topologically distinct candidate equations, constructs a continuous dominance weight field across parameter space, and discovers analytic evolution laws governing mechanism succession---with optional monotonicity constraints encoding irreversible physics. Across four unrelated systems (elastomer mechanics, pool boiling, galactic dynamics, and droplet evaporation), BIC consistently identifies $K^*{=}3$ independent governing topologies. Applied to 163 SPARC galaxies (3,269 spatially resolved measurements), the framework autonomously recovers three gravitational laws whose coexistence provides evidence against the single-universal-acceleration hypothesis of MOND ($p<10^{-34}$). In engineering applications, multi-law weighted prediction reduces error by 67-72\% over single-equation baselines while retaining full interpretability. By establishing the determination of $K^*$ as the zeroth step of scientific discovery-prior to and independent of equation search---this work opens a direction orthogonal to existing symbolic regression: not finding better equations, but mapping the ecology of mechanisms that govern complex systems.

Introduction

Symbolic regression and other equation-discovery approaches generally assume one governing law, an assumption that fails when mechanisms change across conditions or locations. The paper proposes determining the number of independent mechanisms before searching for equations.

Introduction

  • Existing symbolic regression fits one equation with fixed topology, although complex systems can switch mechanisms across conditions, locations, and evolutionary stages.The introduction notes that a single equation can fail to represent these systems even when its parameters are tuned.Source: Introduction · p. 1
  • Before searching for equations, the framework argues, researchers must determine how many topologically independent mechanisms govern an unknown system.Without this “zeroth step,” existing methods may find local optima without mapping the complete mechanism ecology.Source: Introduction · p. 2
  • Physical Law Ecology treats K* as a quantity to discover and views distinct equation topologies as mechanisms competing for dominance in parameter space.The paper presents its algorithmic architecture, theoretical foundations, and cross-domain empirical validation.Source: Introduction · p. 2
  • The framework combines automatic K* determination via BIC, continuous dominance weights, and interpretable closed-form equations, unlike several existing multi-regime approaches.Those approaches may require regime counts or boundaries, use a pre-specified model set, or output a single best-fit expression.Source: Introduction · p. 2

1 Framework overview

Physical Law Ecology maps multiple equations coexisting across space and conditions, then models how their dominance changes. Its pipeline mines distinct equation topologies, constructs continuous weights, learns conditional evolution rules, and uses them for prediction.

1 Framework overview

  • Candle flames illustrate spatial coexistence: different regions have distinct mechanisms, while their local dominance weights change as the candle burns shorter.The figure describes radiation weakening and conduction strengthening at a given location as the candle shortens.Source: §1 · p. 2, Figure 1 · p. 3
  • Three parallel symbolic-regression searches—global, per-condition, and per-spatial-block—mine candidate equations, which are deduplicated by topology into a shared pool.The pool retains the structurally unique equations with the best global fit to cover different regimes and regions.Source: §1 · p. 3
  • For each spatial-condition cell, equation goodness-of-fit is normalized into continuous weights that quantify each equation’s dominance across parameter space.These weights avoid hard discrete assignments and represent gradual mechanism transitions.Source: §1 · pp. 3–4
  • Secondary symbolic regression fits analytic evolution laws to the weight trajectories; optional monotonicity constraints encode trend directions for irreversible processes.The constraints are intended to ensure physical plausibility when mechanisms evolve irreversibly.Source: §1 · pp. 3–4
  • The framework interpolates or extrapolates weights to unseen conditions, then combines coexisting equations through weighted fusion for prediction.Its outputs comprise an equation pool, a dominance field, and evolution laws.Source: §1 · pp. 3–4

2 Elastomer multi-rate tension: automatic discovery of rate-sensitivity structural differentiation

Elastomer tests show that similar rate-dependent stress–strain curves can conceal different mechanism-switching patterns. The framework identifies distinct constitutive laws and predicts held-out rates more accurately than the reported baselines.

2.1 Data and evaluation design

  • Across five polymers, three strain rates, and 4,206 valid measurements, leave-one-rate-out validation trained on two rates and predicted the unseen third.The dataset comprised 45 stress–strain curves measured at 8, 16, and 100 mm/min.Source: §2.1 · p. 4

2.2 Automatic multi-constitutive equation discovery and mechanism differentiation

  • The framework discovered three topologically distinct constitutive equations from data, with forms aligned to the classical polymer constitutive framework.The paper describes these as covering core rubber-response mechanisms, without prior equation assumptions.Source: §2.2 · p. 4

2.3 Prediction performance quantification

  • 0.613 MPa RMSE, 72.1% better than gradient boosting (2.198 MPa) and 66.9% better than global single-law SR (1.850 MPa) in leave-one-rate-out tests.The comparison reports prediction performance for the framework across held-out strain rates.Source: Figure 3 · p. 6, §2.3 · p. 7

2.2 Automatic multi-constitutive equation discovery and mechanism differentiation

  • At 10–30% strain and 100 mm/min, NR60’s linear-elasticity weight reaches 0.83; at 8 mm/min, its three equations coexist at roughly 0.33 each.The high-rate switch from power-law to linear topology is not visible from stress–strain curve shape alone.Source: §2.2 · p. 7
  • At 10–30% strain, NR60’s linear weight rises from 0.33 to 0.83, while EPDM60, NEO60, and CCP60 remain within 0.33–0.46.The paper associates this contrast with equation replacement in crystallizable materials versus balanced coexistence in non-crystallizable materials.Source: §2.2 · p. 7

3 Pool boiling heat transfer: phase transition identification under irreversibility constraints

Pool-boiling analysis identifies distinct heat-transfer laws and shows how their dominance varies with superheat and heater position. A monotonicity constraint encodes irreversible phase succession and improves prediction in held-out conditions.

3.3 Prediction performance and physical insight

  • 25.82 kW/m2 RMSE: 67% better than conventional fifth-order polynomial fitting (61.45 kW/m2) and 20% better than unconstrained ecology (32.21 kW/m2).The constrained result approaches gradient boosting’s 20.64 kW/m2 RMSE while retaining physical interpretability.Source: Figure 4 · p. 8, §3.3 · p. 10

3.1 Data and validation scheme

  • Across 18 conditions and 1,548 measurements, the framework was tested by withholding the most complex conditions from training for prediction.The conditions varied mass flux density, subcooling, and heater position.Source: §3.1 · p. 9
  • The framework identifies three topologically independent heat-transfer equations: strongly nonlinear nucleate boiling, nearly linear transition, and weakly nonlinear film boiling.Their reported exponents are approximately 2.5 for nucleate boiling and 0.33 for film boiling.Source: §3.1 · p. 9

3.2 Physical enabling through monotonicity constraints

  • A monotonicity constraint makes film-boiling weight non-decreasing with superheat, encoding the stated irreversibility of boiling phase transitions.Without the constraint, the fitted film-boiling weight oscillates with local decreases in the 30–55 K range.Source: §3.2 · p. 9
  • At the same 25–45 K superheat, Lower is nucleate-dominated (weight 0.45) while Upper is film-dominated (weight 0.60).The paper attributes the spatial difference to greater liquid-replenishment resistance and earlier film formation at the upper surface.Source: §3.2 · p. 9

4 SPARC galaxy dynamics: deepening understanding of the radial acceleration universal law

The SPARC analysis finds three topologically distinct gravitational equations whose dominance varies with galaxy composition and galactic radius. Robustness tests and links to prior observations support interpreting RAR variation as physical rather than random measurement noise.

4.1 Background: the core conflict between two theories

  • MOND treats the radial acceleration relation as a universal curve with fixed a0, whereas ΛCDM predicts variation with dark-matter halo structure; prior tests of non-universality did not determine how many laws coexist.The paper frames identifying the number and dominance of mechanisms as a question preceding single-equation testing.Source: §4.1 · p. 10

4.2 Data design and analysis mapping

  • The analysis maps 163 SPARC galaxies and 3,269 observations using gas fraction as the composition axis, baryonic acceleration as the regime axis, and observed acceleration as the target.It uses no dark-matter model, gravitational theory, or galaxy-classification prior.Source: §4.2 · pp. 10–12

4.3 Core layered physical discoveries

  • Three distinct equations describe near-Newtonian, square-root transition, and constant-floor regimes, and cannot be reduced to one curve by simple parameter scaling.Their log–log slopes are 0.94, 0.5, and 0, respectively.Source: §4.3 · p. 12
  • Dominance shifts smoothly with gas fraction: gas-poor galaxies are mostly Newtonian and transition laws, intermediate systems share all three, and gas-rich dwarfs are mostly floor-law dominated.At fixed intermediate gas fraction, Newtonian weight recovers near galaxy centres while the transition law dominates outskirts.Source: §4.3 · pp. 12–13
  • BIC selects K*=3, while RAR residuals correlate negatively with gas fraction (r = −0.42, p < 10−34), disfavoring a single universal law in this sample.The reported characteristic acceleration scale is about 2.2-fold lower for gas-rich dwarfs than for gas-poor galaxies.Source: §4.3 · p. 13

4.4 Triple robustness verification

  • Distance-quality filtering, leave-one-galaxy-out validation, and ten half-sample analyses support the persistence of galaxy-type RAR variation across samples and against measurement-error concerns.Leave-one-galaxy-out RMSE improved 2.8% overall and 7.3% for gas-rich dwarfs relative to a single universal curve.Source: §4.4 · p. 14

4.5 Unified explanation of prior observational findings through three-equation ecology

  • The framework interprets prior findings—including low-acceleration constant-like fits, redshift-varying a0, residual correlations, and cluster deviations—as consequences of differing law dominance across galaxy conditions.The authors connect these patterns to differences in galaxy properties and dark-matter halo structure.Source: §4.5 · p. 14

5 Infrared thermography droplet evaporation: experimental validation

Infrared thermography reveals three coexisting evaporation mechanisms and their transitions across temperature, ethanol concentration, and time. Leave-one-concentration-out validation shows that the learned multi-equation ecology predicts held-out radial temperature profiles more accurately than single-equation fitting.

Experimental design

  • Experiments span 24 conditions across four substrate temperatures and six ethanol fractions, using infrared imaging to extract radial temperature profiles.The study asks whether mechanism transitions can be detected from non-invasive temperature fields without presupposing an evaporation theory.Source: §5 · p. 15

Ecological characterization

  • BIC selects three thermal-transport topologies whose weights succeed from conduction early, to Marangoni convection mid-process, to contact-line cooling late.The equations represent radial conduction, a flattened profile from convective mixing, and off-centre cooling associated with contact-line evaporation.Source: §5 · p. 15

Finding 1: Automatic detection of contact-line pinning–depinning transition

  • At 70°C, each additional 10% ethanol advances depinning by 3.2% of evaporation lifetime (R2 = 0.92), as ethanol reduces surface tension and weakens pinning.At 50°C, the cooling peak remains central through about 80% of evaporation before jumping outward at t/ttotal ≈ 0.82, marking the CCR-to-CCA transition without contact-angle or radius measurements.Source: §5 · p. 15

Finding 2: Temperature–concentration effective window of solutal Marangoni

  • At 50°C, solutal Marangoni enhancement is effective only at 10–20% ethanol (Δnmax = +1.09), while at 70°C it rises across 5–50% to +2.18.The proposed explanation is that hotter substrates sustain edge evaporation and the concentration gradient driving solutal Marangoni flow.Source: §5 · p. 16

Finding 3: Evaporation scaling-law deviation quantifies convective enhancement factor

  • Pure water yields α = −0.96, near the diffusion limit, while ethanol-containing mixtures yield α = −1.5 to −1.8 (R2 = 0.98), corresponding to 50–80% enhancement over that limit.The authors propose measuring the scaling relation between evaporation lifetime and substrate temperature as a criterion for identifying the dominant mass-transfer mechanism.Source: §5 · p. 16

Prediction validation

  • 0.70°C overall RMSE, a 14% improvement over single-equation fitting (0.81°C), validates prediction at held-out ethanol concentrations.Improvement reaches 20–30% at intermediate concentrations, where conduction and Marangoni weights are comparable; performance is similar at pure-water and high-concentration endpoints.Source: §5 · p. 16

Universality diagnostic criterion: K∗as the zeroth step of physical discovery

The proposed universality diagnostic makes mechanism number K* the first quantity determined before equation search. BIC-selected K* distinguishes data supporting one topology from evidence of multiple functionally distinct laws.

Universality diagnostic criterion

  • Determine K* by selecting the optimal equation-pool capacity via BIC before searching for governing equations.K* = 1 supports a single topological equation; K* > 1 is quantitative evidence against the universality hypothesis and motivates ecological mapping.Source: Universality diagnostic criterion:… · p. 16
  • The criterion reframes whether a physical relationship is universal as a computable test rather than a qualitative debate.Source: Universality diagnostic criterion:… · p. 16

Scope

Discussion

The discussion presents Physical Law Ecology as a shift from fitting one equation to mapping multiple mechanisms, their dominance, and succession. It contrasts this approach with adaptive and black-box models, while noting limits in operator coverage, extrapolation, and model coverage.

Discussion

  • BIC determination of K* changes the discovery starting point from optimizing a presumed single law to mapping the system’s mechanism ecology.When K* > 1, the framework produces multiple equations, their spatiotemporal dominance weights, and their succession; at K* = 1, it reduces to classical symbolic regression.Source: Discussion · p. 18

Fundamental distinction from parameter-adaptive methods

  • Unlike parameter-adaptive methods that keep equation topology fixed, the framework mines heterogeneous topologies to represent qualitative mechanism changes.Source: Discussion · p. 18

Core distinction from mixture-of-experts black-box models

  • Mixture-of-experts models partition data with black-box networks, whereas this framework outputs interpretable equations and continuous mechanism weights.Source: Discussion · p. 18

Note on baseline comparisons

  • Prediction-accuracy comparisons with single-law symbolic regression are not direct, because the methods target different tasks; manual partitioning plus single-law fitting serves as the closest proxy.The proxy does not automatically determine K*, quantify continuous weight evolution, or provide cross-regime adaptive prediction.Source: Discussion · pp. 18–19

Necessity of ConciseSearch over general-purpose SR engines

  • ConciseSearch uses topology-hash deduplication to supply structurally distinct equations; in an elastomer ablation, PySR’s top five included four equations sharing a power-law topology.The authors describe topological diversity as a prerequisite for meaningful weight fields and evolution laws.Source: Discussion · p. 19

Applicability and recommended workflow

  • The recommended workflow determines K* first, then uses classical symbolic regression for K* = 1 or maps the equation pool, dominance field, and evolution laws for K* > 1.The four validation cases represent condition-induced gradients, irreversible phase transitions, global mechanism heterogeneity, and spatiotemporal evolution.Source: Discussion · p. 19

Limitations

  • The framework’s current operator vocabulary omits trigonometric and special functions, extrapolation degrades far from training conditions, and coverage is typically 85%–95%.Source: Discussion · p. 19

Future directions

  • Future work targets broader cross-domain K* benchmarks, mechanistic studies of complex systems, and a dynamical theory of law ecology.Source: Discussion · p. 19

Methods

The methods define a probabilistic dominance field, interpret coexisting mechanisms through ecological and phase-transition concepts, and use BIC to select equation-pool size. They also describe the data-to-output pipeline and the ConciseSearch engine used to mine distinct equation topologies.

Bayesian formulation: dominance field as posterior probability field

  • Under Gaussian noise and equal equation priors, each dominance weight represents the posterior probability that its equation is the true generative model at that location and condition.A weight near one indicates single-mechanism dominance; uniformly distributed weights indicate balanced coexistence.Source: Bayesian formulation: dominance field… · p. 20

Ecological isomorphism: from metaphor to rigorous mathematical correspondence

  • The ecological isomorphism maps equations to species, parameter space to habitat, and dominance weights to abundance; local coexistence is limited by data dimensionality.The framework relates this dimensionality limit to the ecological competitive exclusion principle.Source: Ecological isomorphism: from metaphor… · p. 20

Phase-transition analogy: order parameter characterization of mechanism transitions

  • The order parameter ψ = w1 − w2 distinguishes continuous second-order transitions from abrupt first-order transitions; monotonicity represents one-way succession, including nucleate-to-film boiling.The monotonicity constraint is interpreted through supercooling/superheating asymmetry.Source: Phase-transition analogy: order… · pp. 20–21

Information-theoretic model selection: automatic determination of equation pool capacity

  • BIC selected K*=3 for elastomer, pool boiling, SPARC, and droplet evaporation, balancing model fit against structural complexity.The optimal pool size is the K value minimizing the ecological description length.Source: Information-theoretic model selection:… · p. 21

Algorithm overview

  • The pipeline takes spatial location, condition, features, and target observations as input and outputs an equation pool, dominance weight field, and evolution laws.Source: Algorithm overview · p. 21

ConciseSearch symbolic regression engine

  • ConciseSearch de-duplicates equations by topology, schedules complexity penalties, and refines constants each generation using L-BFGS.Experiments use population 200–300, 40–60 generations, maximum tree depth 4, complexity cap 10–12 nodes, and random seed 42.Source: ConciseSearch symbolic regression engine · pp. 21, 23

Multi-path equation pool construction strategy

The section describes how the framework builds a heterogeneous equation pool, computes dominance weights, and mines conditional evolution laws. It also reports synthetic validation and summarizes dataset, baseline, and statistical-test settings.

Multi-path equation pool construction strategy

  • Global, per-condition, and per-spatial-block symbolic-regression searches capture dominant, minority-regime, and locally specific laws before topologically distinct equations are retained in a shared pool.The retained top-K equations are ranked by global goodness-of-fit.Source: Multi-path equation pool construction… · p. 23

Continuous dominance weight field computation

  • For each spatial-block and condition-window cell, negative R² values are clipped to zero and the remaining equation scores are normalized into dominance weights.The weight data are concatenated into a continuous, smooth field across parameter space.Source: Continuous dominance weight field… · p. 23

Evolution law mining and monotonicity constraint implementation

  • Secondary symbolic regression models each block’s weight–condition sequence; trend verification rejects candidates with counter-trend anomalous points exceeding 20%.Expected trend direction is prescribed from physical priors to accommodate irreversible processes.Source: Evolution law mining and monotonicity… · p. 23

Ablation and synthetic validation

  • With 2% Gaussian noise across 2,000 synthetic samples, the framework recovered all topologies, with square-root coefficient error <6%, linear error <16%, and full-domain mechanism coverage 100%.Sweeping K = 3 to 8 and manual settings within one of the BIC optimum showed robustness to pool capacity.Source: Ablation and synthetic validation · p. 23

Dataset sources and baseline method settings

  • Elastomer data follow ASTM D412-16, pool-boiling data are high-precision laboratory measurements, and SPARC observations require quality flag ≤2; baselines include single-law SR, boosting, and polynomial fitting.The elastomer data come from Dryad and the SPARC data from a public astronomical database.Source: Dataset sources and baseline method… · pp. 23–24

Statistical test methods

  • Galaxy residual analyses use two-sided Spearman rank correlations and Kolmogorov–Smirnov tests, with significance judged at p < 0.05.Source: Statistical test methods · p. 24

Code availability

The authors provide the framework and engine code alongside materials for reproducing the application cases.

Code availability

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