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Constrained weak identification of Allen--Cahn free energies with surface-tension calibration

Soobeen Jung, Hyunju Kim

arXiv:2609.18403v1math.NAcs.CEcs.NAnlin.PSphysics.comp-ph

TL;DR

Allen–Cahn trajectories determine the effective force but not the free-energy potential and interfacial scale separately. STAC estimates the force with weak moments, imposes a positive Bernstein structure, and uses supplied surface tension for calibration. In shared noisy records, positivity with ridge regularization removes structural violations and reduces mean force error, while degree choice and approximation bias remain relevant.

  • Problem

    Resolved Allen–Cahn trajectories determine G = F′/ε² but cannot distinguish the free-energy potential F and interfacial scale ε individually.

    Source: §1 · p. 2

  • Method

    STAC estimates the effective force from space–time weak moments with positive Bernstein coefficients, then calibrates its scale using independently supplied surface tension.

    Source: §1 · p. 3, §3.1 · p. 8

  • Results

    8.70% to 5.66% mean force error across 80 shared noisy records with positivity and ridge regularization, which removes all six degree-five structural violations.

    Source: §5 · p. 31

  • Takeaways & Limitations

    Trajectories supply the effective force while the independent surface-tension datum supplies the physical scale, making their distinct information roles explicit.

    Source: §5 · p. 31

    The implementation prescribes constitutive degree because tested selection procedures were unreliable, and positive Bernstein constraints are sufficient but restrictive.

    Source: §3.1 · p. 9, §4.5 · p. 25

Abstract

from arXiv · show

Phase-field trajectories governed by the Allen--Cahn equation determine the effective force $G=F'/\eps^2$, but not the free-energy potential $F$ and interfacial scale $\eps$ separately. We introduce surface-tension-calibrated Allen--Cahn identification (STAC), a two-step method that resolves this ambiguity using an independently supplied surface tension. First, space--time weak moments produce a linear inverse problem without differentiating the measured field. A Bernstein representation with positive coefficients enforces two stable pure phases and a single central energy maximum. Second, the surface tension fixes the remaining scale algebraically. We prove scale equivalence of the forward model, uniqueness of the calibrated pair within this class, and finite perturbation bounds that remain valid when the potential vanishes at the pure phases. Experiments with resolved synthetic phase fields separate the effects of the structural constraint, ridge regularization, and constitutive approximation error. Across 80 shared noisy records, unconstrained degree-five regression produces six potentials with an incorrect well structure. Positivity combined with ridge regularization eliminates these violations and reduces the mean force error from 8.70\% to 5.66\%. Lower-degree models are nevertheless more accurate in some tests, and an admissible law outside the positive Bernstein family shows how the constraint can introduce approximation bias. STAC thus recovers a structurally admissible, physically scaled potential while making explicit which information comes from the observed dynamics and which comes from the independent surface-tension datum.

1. Introduction

Allen–Cahn trajectories identify the effective force but cannot separate the free-energy potential from its interfacial scale. STAC combines weak-moment estimation with an independently supplied surface tension to recover a structurally constrained, physically scaled potential.

1. Introduction

  • Trajectories determine G = F′/ε², but the forward model is unchanged when ε and F are rescaled together, so an independent energy datum is needed.More accurate or denser trajectories cannot resolve this scale ambiguity.Source: §1 · p. 2
  • Prior approaches include PDE-constrained optimization, variational identification, sparse regression, and neural methods, but trajectory data alone cannot recover a scale absent from the dynamics.The problem also requires a shape-preserving estimate and a separate physical-scale choice.Source: §1 · pp. 2–3
  • STAC estimates the effective force from space–time weak moments without differentiating measurements, then uses supplied surface tension to select the physical scale.Positive Bernstein coefficients enforce the prescribed symmetric double-well structure; surface tension is not inferred from the trajectory.Source: §1 · pp. 2–3
  • Across 80 shared noisy records, positivity with ridge regularization removes six degree-five structural violations and lowers mean force error from 8.70% to 5.66%.The comparisons also report cases where degree two is more accurate and an outside-family law for which the constraint adds bias.Source: §1 · pp. 3–4
  • The numerical study uses resolved synthetic observations and prescribes constitutive degree because tested selection procedures were not reliable across all laws and noise levels.The constraint can add bias when the admissible law lies outside its family.Source: §1 · p. 4

2. Identifiability and surface-tension calibration

The Allen–Cahn dynamics identify a normalized force, not an absolute potential and interface scale. A supplied, consistently normalized surface tension selects the scale, with finite bounds describing calibration sensitivity and its conditions.

2.1. Energy, mobility, and the identifiable force

  • The Allen–Cahn operator depends on F and ε only through G = F′/ε², so rescaling them as (sε, s²F) leaves the operator and its solution sets unchanged.This equivalence concerns forces equal throughout the phase interval, not complete identifiability from one trajectory; limited phase coverage or ill-conditioned moments can leave additional forces indistinguishable.Source: §2.1 · p. 5

2.2. Calibration from interfacial energy

  • For a nonnegative normalized potential with finite positive tension, each scale class has exactly one member matching a supplied positive surface tension.The supplied datum must use the calibration formula’s energy normalization; otherwise the recovered scale is incorrect even if the trajectory agrees.Source: §2.2 · p. 6, §2.3 · p. 8
  • Interface width is unchanged by the scale transformation and determines ε only after the potential’s amplitude and shape are fixed.For curved, evolving interfaces, the planar interface formula is approximate, although the equation’s scale invariance remains exact.Source: §2.2 · p. 6

2.3. Finite calibration stability and endpoint sensitivity

  • Finite calibration bounds cover simultaneous perturbations of the primitive and supplied tension, including potentials that vanish at the pure phases.The inverse-scale stability additionally requires the primitive’s calibration integral to be bounded away from zero; the stated interval can be conservative.Source: §2.3 · pp. 6–7
  • With the primitive fixed, a relative change in supplied tension changes ε proportionally and F by the square of that factor.Errors in the recovered force, calibrated potential, and scale are distinct; the tension datum must match the calibration normalization.Source: §2.3 · p. 8

3. Weak identification and constrained estimation

The section builds a constrained weak estimator for the effective force, then calibrates the potential and interfacial scale using supplied surface tension. Its positive Bernstein family guarantees the prescribed double-well structure, while the analysis distinguishes structural constraints, ridge regularization, and approximation or observation error.

3. Weak identification and constrained estimation

  • The estimator first solves a constrained weak problem for the effective force, then uses supplied surface tension to determine the potential and interfacial scale.The weak solve is followed by integration of the primitive and algebraic calibration.Source: §3 · p. 8, §3.3 · p. 13

3.1. A positive constitutive family

  • Positive Bernstein coefficients ensure two stable pure phases and one central energy maximum, but this sufficient condition excludes some physically admissible laws.The positive lower bound also makes the curvatures at the pure phases and central maximum strict.Source: §3.1 · p. 9
  • At degree two the nonnegative-cone approximation floor is 37.68%, versus 5.55% at degree five; the unconstrained spaces represent the law exactly.These evaluation-only errors measure approximation capacity, not recovery from trajectory data.Source: §3.1 · pp. 9–10, Figure 1 · p. 10

3.2. Space–time moments and their discrete evaluation

  • Space–time distributional moments yield a weak identification system without classical derivatives of the observed field, but discretization and nonlinear noisy features are not made exact.The test functions use compact interior supports, and the discrete design retains derivative margins and excludes bulk-phase centers.Source: §3.2 · p. 11

3.3. Constrained estimate, shrinkage, and conditional stability

  • The constrained solution is a Q-metric projection that cannot worsen the moment residual against a feasible reference; ridge separately shifts design eigenvalues and can add bias.The constraint changes feasible directions, whereas ridge changes the Hessian; stability bounds require common-row and feasibility conditions.Source: §3.3 · pp. 12–14

3.4. Systematic and statistical residuals

  • Approximation and forward-discretization residuals are systematic, while averaging nonlinear-feature estimates can reduce variance without removing bias.The reported ensembles do not establish calibrated coefficient intervals or consistency at fixed nonzero noise.Source: §3.4 · pp. 14–15

3.5. Experimental variants and comparison design

  • The comparison design varies ridge strength with equal tuning budgets and contrasts analytic-derivative and smoothed pointwise controls against discrete-derivative selection.The controls use positive coefficients and shared validation moments, while the finite-difference control fixes degree two.Source: §3.5 · p. 15

4. Numerical results

The numerical study uses synthetic phase fields to compare recovery across laws, geometries, noise levels, and estimation protocols. Paired methods share observations and noise realizations, while bootstrap summaries are based on only ten seed clusters.

4.1. Test problems and comparison protocol

  • The principal benchmark uses circular and nonconvex initial fields with four symmetric potentials, including polynomial and nonpolynomial laws.The exponential and rational laws lie outside every finite polynomial space, separating approximation error from observation error.Source: §4.1 · p. 16
  • The principal study comprises eight clean records and 80 records at each of noise standard deviations 0.01 and 0.03, using ten paired seeds per law–geometry pair.The setup uses a unit-square domain, q=1, ε=0.03, a 96 grid, and 101 generated fields.Source: §4.1 · p. 16
  • Paired comparisons use identical observations and noise realizations, but bootstrap intervals from ten seed clusters do not establish nominal 95% coverage.The intervals summarize repeated noise on the tested cases; deterministic bounds are reported separately.Source: §4.1 · pp. 16, 18
  • A separate 2562 dataset supplies sixth-degree-law observations across circular, nonconvex, and inclusion geometries for neural comparisons, not the four-law principal ensemble.The inclusion retains inner and outer interfaces over the displayed interval.Source: §4.1 · p. 16, Figure 2 · p. 17
  • The force-error measure evolves recovered and reference laws from the same clean frame over a fixed horizon; calibrated potential and scale errors also depend on supplied tension.Structural violations and failed calibrations remain represented in reported counts, with failed calibration errors left undefined.Source: §4.1 · p. 18

4.2. Effects of positivity and ridge regularization

On 80 principal records at noise 0.03, positivity with ridge regularization gives the lowest reported degree-five mean force error and removes structural violations. Comparisons across degrees and against finite-difference regression show that these gains depend on the law and test setting.

4.2. Effects of positivity and ridge regularization

  • 5.66% mean force error for degree-five positivity plus ridge, versus 8.70% for unconstrained least squares on the 80 principal records.The comparison separates positivity (6.20%) and ridge alone (7.29%) from their combination.Source: §4.2 · p. 18, Table 1 · p. 19
  • Unconstrained degree-five regression violates the prescribed well structure in 6 of 80 records; both constrained branches retain it in every record.Ridge alone reduces violations to four but does not eliminate them; the constraint is active in 52 unpenalized and 25 ridge estimates.Source: §4.2 · p. 18
  • The unconstrained example adds force zeros near u = ±0.0835 and turns the central energy maximum into a local minimum, making the mixed state locally metastable.The force curve remains close to the reference, so small average trajectory error need not reveal this local stability error.Source: §4.2 · p. 19, Figure 4 · p. 21
  • Adding positivity to ridge changes mean error by −1.63 pp, while adding ridge to positivity changes it by −0.54 pp on principal records.The combined estimate differs from fixed degree-two STAC by +0.01 pp; on fresh seeds, degree-five ridge STAC is 6.67% versus 5.65% at degree two.Source: §4.2 · pp. 18–19, Figure 3 · p. 20
  • For the degree-three law, degree-five ridge STAC reduces error by 7.09 pp; for the other three laws, additional coefficients increase error by 1.13–3.40 pp.The degree-two approximation floor is 6.45% for the degree-three law and at most 0.60% for the other three.Source: §4.2 · p. 20, Table 2 · p. 21
  • At degree two, positivity is inactive and ridge changes little; the −7.43 pp improvement over smoothed FD4 concerns weak versus pointwise formulations, not positivity.The ordering reverses on separate clean 2562 controls, where direct FD4 is more accurate; neither result supports a universal ranking.Source: §4.2 · p. 20
  • Final-field error e_u stays at 0.111–0.125% across fixed weak branches, but is less discriminating than force error because trajectories visit limited phase values over a short horizon.Agreement in e_u complements force-error and local-stability checks.Source: §4.2 · pp. 20, 22

4.3. Replication and the effect of constitutive degree

Replication supports separate effects of positivity and ridge regularization, but does not establish that the richer degree-five family is preferable. Supplementary clean tests show that recovery can worsen with degree even as approximation improves.

4.3. Replication and the effect of constitutive degree

  • Across four conditions in the 40-record replication ensemble, constrained estimates have smaller mean error than unconstrained estimates, while structural violations rise from 1 to 12 of 40 as observations become less informative.Some interval conclusions depend on construction; for example, the short-record contrast at noise 0.01 has a cluster-t interval containing zero.Source: §4.3 · p. 22
  • The combined degree-five estimator has the lowest mean error among degree-five branches in every condition, but degree-two STAC is more accurate at noise 0.03 for both record lengths.Only the constrained branches avoid all structural violations; replication supports positivity and regularization separately without establishing that the richer family is preferable.Source: §4.3 · p. 22
  • In clean circular tests, exponential-force approximation improves with degree while trajectory recovery is most accurate at degree four and deteriorates at degree five.The increasing condition number exposes poorly determined coefficient combinations; these supplementary tests use an earlier normal-equation implementation, not the principal QR/SVD solver.Source: §4.3 · pp. 22–23, Figure 5 · p. 23
  • For the sixth-degree law, spatial refinement reduces clean force error from 0.0163% at N = 128 to 5.3 × 10^-5% at N = 512.This resolved clean test indicates a small discretization contribution, but does not separate discretization and observation errors for every law or establish consistency at fixed nonzero noise.Source: §4.3 · p. 23, Figure 5 · p. 23

4.4. Recovery when the true law is outside the positive family

An admissible law outside the positive Bernstein cone tests the structural restriction's approximation cost. Its effect on recovery varies with degree and observation noise, so positivity guarantees structure but not uniformly accurate force recovery.

4.4. Recovery when the true law is outside the positive family

  • At degree two, positivity raises clean force error from 0.45% to 60.82% and noise-0.01 error from 2.64% to 61.31%, although unconstrained estimates are structurally admissible in these 22 records.The constrained errors exceed the 37.68% function-space approximation floor because weak moments weight visited phase values differently from the uniform phase-space norm.Source: §4.4 · p. 24
  • At degree five and noise 0.01, constrained error is 6.93% versus 13.15% unconstrained, while 16 of 20 unconstrained estimates violate the prescribed structure.The 5.55% approximation floor is low, and four constrained coefficients reach the lower bound in every record, suppressing large unconstrained oscillations.Source: §4.4 · p. 24
  • At noise 0.03, degree-five constrained error rises to 27.89%, exceeding the unconstrained error of 10.00% despite the same approximation floor.Feature bias and changes in the observation-dependent mask alter the weak system as noise increases.Source: §4.4 · pp. 24–25
  • At degree two and noise 0.03, both estimates coincide because unconstrained coefficients are already positive, despite a force error of 77.80%.An inactive constraint therefore does not show that the candidate family accurately describes the law.Source: §4.4 · p. 25
  • Positivity guarantees the declared well structure, but recovery depends jointly on degree, trajectory, and observation error; the reference-based approximation floor is unavailable when the law is unknown.The tested law has the required symmetric double well but lies outside the nonnegative cone at degrees two and five.Source: §4.4 · pp. 23–25

4.5. Limits of observation-based degree selection

Observation-based degree selection is unreliable across laws and noise levels, so the implementation prescribes degree. The results show that neither pooled accuracy nor feature-mean correction resolves selection consistently.

4.5. Limits of observation-based degree selection

  • Degree selection remains unresolved: pooled principal and fresh-seed results favor low degree, while the degree-three law benefits from a richer family.A small validation residual is insufficient to identify the most accurate representation under these conditions.Source: §4.5 · p. 25
  • 4.81% to 2.90% at noise 0.01, but 5.65% to 6.78% at noise 0.03: disjoint validation blocks improve mean force error only inconsistently.For the degree-three law, the rule selects degree one in all 20 high-noise records, despite a law-by-law benefit from additional coefficients.Source: §4.5 · p. 25
  • 10.72% to 8.19%: Hermite correction reduces degree-two mean force error, but selected-degree error changes by only −0.05 pp, with interval [−1.30, +1.28].The limited selection improvement suggests feature-mean bias is only one source of error; the correction uses estimated variance and does not imply unbiased coefficients or validation scores.Source: §4.5 · p. 25
  • The present implementation prescribes degree; this is a practical conclusion about tested criteria and observations, not proof that data cannot reveal constitutive complexity.Reliable adaptive degree selection requires further study; numerical rank, support sensitivity, and phase coverage remain useful diagnostics.Source: §4.5 · p. 25
  • Figure 6 reports mean relative force error across six observation conditions and includes structurally invalid unconstrained estimates.The selected STAC branch uses the exploratory observation-based degree and support rule.Source: Figure 6 · p. 26

4.6. Effect of observation quality

Coarser observations and thinner interfaces increase quantitative recovery error, while structural constraints preserve the prescribed well shape. Errors also depend on phase coverage, noise, and the supplied observation regime.

4.6. Effect of observation quality

  • 11.2% and 13.5%: coarse observations and a thinner interface raise fixed degree-two STAC force error, versus 2.7% in the reference configuration.Both changes reduce samples across the transition layer; correlated noise also weakens averaging because neighboring fluctuations are not independent.Source: §4.6 · p. 26
  • 41 and 55 of 120 records: unconstrained degrees two and five violate the well structure; degree five also fails calibration in three records, while constrained estimates have no violations.The table summarizes constraint activation and structural violations over the 120 observation-stress records.Source: §4.6 · p. 26, Table 5 · p. 27
  • Approximately 13% and 25–33%: fixed-degree STAC force errors at noise 0.05 and 0.10, respectively, despite retaining admissibility.These are limits in quantitative recovery, not evidence that constraints recover information absent from the record.Source: §4.6 · p. 27
  • 6.44% from 8.70% without constraints and 4.77% from 6.20% with constraints: phase-histogram weighting lowers the principal degree-five mean errors.The largest discrepancies are partly concentrated in less-sampled phase values, so trajectory-weighted error alone can conceal limitations when the law is used with a different initial field.Source: §4.6 · p. 27

4.7. Comparison with neural constitutive estimates

On identical observations in the reported neural comparison, the unconstrained neural forces fail the prescribed double-well conditions, while projection onto a positive degree-two family changes both admissibility and constitutive error. The comparison also reports implementation-specific timing and training-scope caveats.

4.7. Comparison with neural constitutive estimates

  • 2.64–5.45% to 0.200–0.252%: projection reduces SPINN eG across geometries, making it comparable to and sometimes more accurate than STAC.The projected result reflects substantial structural information supplied by postprocessing, not the network alone.Source: §4.7 · p. 27, Table 6 · p. 28
  • The unconstrained neural forces do not satisfy the prescribed double-well conditions in these runs; projection onto the common positive degree-two family enforces them and changes constitutive error.Table 6 distinguishes raw network forces from their projected versions; potential errors are reported after projection.Source: §4.7 · p. 27, Table 6 · p. 28
  • PFWNN has larger errors under the reported training budget.Table 6 uses a separate sixth-degree-law study at noise 0.01, with one measurement realization per geometry and three network initializations.Source: §4.7 · pp. 27–28, Table 6 · p. 28
  • Approximately 1.2–1.3 s: STAC CPU identification time, versus 165–382 s of GPU training for the neural methods in these implementations.The hardware differs, and three initializations on one noise realization quantify optimization variability rather than performance over measurement noise.Source: §4.7 · p. 28

4.8. Interpretation of the calibrated potential and scale

Calibration can yield different error levels for potential shape and interfacial scale, which depends on the calibration integral. The numerical checks show that datum accuracy matters: an inaccurate supplied tension can dominate otherwise accurate force recovery.

4.8. Interpretation of the calibrated potential and scale

  • −0.0615% to −0.5446%: signed degree-two scale bias changes from noise 0.01 to 0.03, while potential shape error can remain larger despite small scale error.The increase is consistent with a squared-noise contribution from nonlinear feature bias, but does not isolate that mechanism.Source: §4.8 · p. 28, Figure 7 · p. 29
  • 0.61–0.93% tension errors from an under-resolved equilibrium calculation increase mean scale and potential errors to 1.24% and 2.30%, respectively.The finest mesh has relative tension errors of (1.7–2.6)×10^-4%, with negligible changes in calibrated errors; both calculations use the reference potential.Source: §4.8 · pp. 28, 30
  • 10% tension coefficient of variation raises root-mean-square scale and potential errors from 0.0557% and 1.45% to 10.3% and 21.1%, respectively.This separate test uses the exploratory observation-selected estimator, a different baseline from fixed-degree STAC; conditional tension draws do not increase independent field realizations.Source: §4.8 · p. 30
  • For fixed calibration integral, a factor s in supplied tension changes scale by s and potential by s^2 exactly; the approximately 0.1% crossover is specific to the selected-estimator test.An application requires a physically matched datum in consistent units and must propagate its uncertainty; the synthetic study infers no material-specific uncertainty.Source: §4.8 · p. 30

5. Conclusions

STAC separates what resolved trajectories identify from the absolute scale supplied by surface tension. Numerical comparisons show that structural constraints and ridge regularization improve degree-five results, while approximation bias and the study's scope limit the conclusions.

5. Conclusions

  • Resolved trajectories determine the effective force G = F'/ε^2 but not F and ε separately; supplied surface tension selects the physical scale, with uniqueness and perturbation bounds characterizing calibration.The bounds account for errors in both the recovered primitive and supplied tension.Source: §5 · p. 31
  • Combining Bernstein positivity with ridge regularization removes all six degree-five structural violations and reduces mean force error from 8.70% to 5.66%.Positivity preserves prescribed phase stability, while ridge regularization controls poorly determined coefficient directions.Source: §5 · p. 31
  • Results are limited to resolved synthetic phase fields, known diffusion, and symmetric equal-depth wells; experimental use also requires uncertainty estimates for measured surface tension.Degree selection, confidence intervals, incomplete observations, and broader potential classes remain open topics.Source: §5 · p. 31
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