Inferring constitutive forces from noisy Allen-Cahn movies with moment equations

1 Introduction

• Prior approaches estimate constitutive relations, states, or laws, but the paper focuses on which estimating-equation errors must be removed before adaptive inversion.The cited routes include trajectory inversion, physics-informed and weak-form models, and sparse equation regression.Source: §1 · p. 1
• Noisy phase values bias nonlinear force features, while instruments built from the same observations correlate with response noise.The method uses Gaussian Hermite polynomials for the first effect and Gaussian recorruption with opposite views for the second.Source: §1 · p. 1
• A temporal balance borrowed from another solver can remain biased even on clean data, so the balance must match the observation source.The paper derives a discrete balance when generator metadata are known and otherwise uses an observation-time weak balance.Source: §1 · p. 2
• For the tested inverse problem, correcting the estimating equation contributes more than adding a more elaborate regularizer.SCCT combines Gaussian polynomial correction, independent-view instruments, and an observation-source-matched balance with a constrained Bernstein inverse.Source: §1 · p. 2
• The experiments isolate noise corrections, test balances across forward generators, compare regularizers on independent laws, and assess transfer to unseen fields.The three-dimensional test distinguishes field accuracy from a topology change that can persist under inaccurate forces.Source: §1 · p. 2

2.1 The identifiable constitutive target

• Trajectories determine the constitutive force target, while an external surface-tension measurement sets the remaining energy scale.A parameter transformation leaves the dynamics unchanged, so the force is inferred first and calibrated afterward.Source: §2.1 · p. 2, §2 · p. 2

2.2 Admissible force and observations

• The admissible force is represented by five positive coefficients, whose family retains two pure-phase minima and one central unstable state.The force and its first derivative are continuous at the endpoints after extension beyond the phase interval.Source: §2.2 · p. 3
• The family excludes asymmetric wells and displaced minima, and sparse sampling of intermediate phase values can leave even an in-family force poorly determined.The admissible model therefore bounds which force shapes the inverse can recover.Source: §2.2 · p. 3
• Inference uses noisy observed phase values, with independent standard Gaussian noise unless stated otherwise; the noise level is supplied or estimated.The noise level is distinct from the downstream surface-tension datum.Source: §2.2 · p. 3

3 Noise-adjusted moment equation and inverse

• The method first corrects systematic errors in the estimating equation, then uses a common positive inverse across regularization choices.External surface-tension calibration is a subsequent energy-scale step, not part of the shared force solve.Source: §3 · p. 3, Figure 1 · p. 4

3.1 Noise-adjusted estimating moments

• Gaussian Hermite-adjusted features remove nonlinear-substitution bias when the variance is correct, while paired recorrupted views cancel the instrument–response cross-noise term under the known iid Gaussian model.Plug-in noise estimates make both cancellations approximate; interpolation and model residuals remain.Source: §3.1 · pp. 3, 5
• Nine overlapping phase instruments overdetermine five force coefficients, but their count alone does not guarantee full rank.The system pairs opposite views and averages artificial perturbation pairs, which reduce artificial-perturbation variability but are not independent replicates.Source: §3.1 · p. 4

3.2 Discrete and observation-time balances

• The temporal balance must match the observation generator: a discrete balance reconstructs internal stages when metadata are known, while an observation-time weak balance uses observed samples.The weak balance avoids reconstructing internal states but retains temporal-quadrature and spatial-discretization errors.Source: §3.2 · p. 5

3.3 Positive inverse and optional learned regularization

• Positive Bernstein inverses of degrees 1, 2, and 4 are blended into a trajectory-dependent prior, with SCCT adapting prior and precision from images and moments.All branches use the same admissible family and constrained solve; reduced images supply context while moments retain observation resolution.Source: Figure 1 · p. 4, §3.3 · pp. 5–6

3.4 Residual control and inverse stability

• The conditional stability bound separates moment residual from prior displacement and tracks how both are amplified by the realized inverse.Positive precision ensures positive definiteness for noisy observations; at zero noise, full column rank is required, and uniform pure-phase equilibria fail this condition.Source: §3.4 · p. 6
• The force-error bound depends on the evaluation-grid Gram matrix, and the same coefficient error can produce different force errors depending on its direction.For out-of-family truth, the error measures distance to a chosen admissible reference and retains an approximation discrepancy.Source: §3.4 · p. 6
• The stability result is deterministic for the observed system; a stochastic accuracy guarantee additionally requires bounding the contributions to the moment residual.Those contributions include observation noise, temporal reconstruction, spatial discretization, and force-family discrepancy.Source: §3.4 · p. 6

4 Experimental design and results

• The numerical study uses synthetic trajectories, including a 61-frame primary trajectory on a 128^2 periodic grid, with Gaussian noise levels η ∈ {0.001, 0.01, 0.03} and clean controls.Reduced-frame studies vary sampling intervals, while the original collection contains 768 training laws and separate 160-law validation and test sets.Source: §4 · p. 7

4.1 Correcting the equation matters more than retraining

• At η = 0.03, removing polynomial correction increases adaptive-map error by 1.40 pp [1.24, 1.56], versus 0.24 pp [0.18, 0.31] when removing views.Across tested noise levels, both corrections become beneficial from η = 0.01 upward; the non-neural smoothness inverse shows the same moderate-noise ordering, locating the improvement in the estimating equation.Source: §4.1 · pp. 8–9, Table 1 · p. 10

4.2 The observation source selects the estimating balance

• At η = 0.01 for ETDRK4 observations, using observation-time weak moments reduces SCCT error from 0.371% to 0.219%, while IMEX observations favor the discrete balance.The preferred balance reverses across the two generators and holds across tested noise levels and regularizers. Sparse sampling raises weak-path errors, and weak-form results also depend on the spatial observation operator; with moments fixed, the adaptation comparison is smaller.Source: §4.2 · pp. 10–12, Table 2 · p. 11

4.3 Learned regularization is secondary after noise adjustment

• On the original 160 test laws, SCCT reduces mean force error from 0.237% for the global mixture to 0.216%, a paired difference of −0.021 pp [−0.033, −0.010].On fresh laws, the original adaptive models retain a small advantage, but common-training precision and structural controls do not establish a general adaptation advantage; the original-versus-refit contrast cannot isolate a training mechanism.Source: §4.3 · pp. 12–13, Figure 7 · p. 13

4.4 Information loss and model departures limit recovery

• Late trajectories without an interface have moment condition numbers of 1.2 × 10^9–10^16 and errors of at least 22%, while interface-retaining trajectories have errors below 6.5%.More independent trajectories reduce global-inverse mean error from 0.208% with one trajectory to 0.128% with two and 0.091% with four. Noise-model misspecification and out-of-family forces also impair recovery, and tested observation-only diagnostics cannot certify estimates from unknown laws.Source: §4.4 · pp. 14–15

4.5 Recovered laws support reuse within that boundary

• For unseen initial fields, inferred forces retain field accuracy, while all four inference methods reproduce 107 of 108 topological events and tilted exact laws reproduce all 108.Continuous-field errors distinguish imposed force perturbations more clearly than event labels; the finite-budget trajectory inverse has larger errors than SCCT, moment-only, and global regularization.Source: §4.5 · pp. 15–16, Table 4 · p. 16
• A 5% increase in supplied σ raises bε by 5% and bF by 10.25% with G held constant, while forecasts remain unchanged.Surface tension calibrates energy rather than the force or forecast scale.Source: §4.5 · p. 17
• Under lognormal surface-tension uncertainty with s = 0.05, mean SCCT potential error was 7.97%, distinct from the deterministic perturbation identity.The experiment used σobs = σ⋆exp(sz −s2/2) with z ∼N(0, 1).Source: §4.5 · p. 17
• Bootstrap intervals covered ε near nominal 90% only in the designed surface-tension-uncertainty setting; on nonpolynomial laws, pointwise force coverage fell to 9%.No norm-ball interval covered the truth for the eight nonpolynomial laws; intervals remain conditional on the assumed family and noise model.Source: §4.5 · p. 17

5 Discussion and conclusions

• The retrained factorial assigns the largest accuracy change to moment corrections, while the IMEX–ETDRK4 crossover shows that temporal balance must follow the data generator.Covariance modeling [20] [21] can address residual dependence beyond these corrections.Source: §5 · p. 17
• The original SCCT checkpoints retain a small fresh-law advantage, but common-training precision maps and structural controls perform close to the global regularizer.Different training histories prevent attributing the advantage to a particular architectural feature; adaptation depends on the inverse and data distribution.Source: §5 · p. 17
• The conditional bound tracks how moment residuals and prior displacement affect the positive inverse, explaining learned regularization’s smaller role in the tested regime.Its role depends on available phase information.Source: §5 · p. 17
• The recovered force forecasts unseen planar fields and three-dimensional evolutions without a new inverse solve, while external surface tension sets the energy scale independently.Transfer is conditional on an informative movie and the assumed force family.Source: §5 · p. 17
• Ring-cage breakup labels can agree across perturbed forces even when their diffuse-interface errors are much larger.Late frames that lose the interface, correlated noise, and out-of-family laws challenge the assumptions; current diagnostic alarms remain insufficient.Source: §5 · p. 17

A Study protocol and statistical units

• Comparisons treat the generating constitutive law as the independent unit, averaging repeated measurements and model variations within each law before law-level resampling.The study overview likewise reports independent test laws rather than trajectories or frames.Source: A Study protocol and statistical units · p. 18, Table A.1 · p. 18

B Moment identities and inverse properties

• Appendix B supplies the algebra for the admissible force, moment corrections, and conditional inverse bound used in the main text.For positive Bernstein coefficients, the force has the prescribed well structure, and the implemented extension restores force outside the wells.Source: B Moment identities and inverse… · p. 19, §B.1 · p. 19

B.1 Admissible family and numerical normalization

• The force family uses positive Bernstein coefficients with a numerical floor, and degree elevation preserves the coefficient floor so convex mixtures remain admissible.Surface-tension calibration uses 96-point Gauss–Legendre quadrature; a 96-to-192-node refinement check changed results by less than 1.5 × 10^-14.Source: §B.1 · p. 19

B.2 Filters and instrument coverage

• Nine overlapping phase instruments overdetermine five coefficients, while temporal and spatial filters reduce variance without guaranteeing rank.Filter supports and observation-time temporal support are implementation settings, not universal optima.Source: §B.2 · p. 19

B.3 Known-generator and observation-time balances

• The known-generator balance follows the observation-producing IMEX stages, but interpolation from six neighboring frames adds a reconstruction residual.Reproducing the time-stepping scheme does not remove that residual.Source: §B.3 · pp. 19–20
• When internal stages are unavailable, the observation-time balance integrates the PDE against a compactly supported temporal test and uses observed windows without internal-stage interpolation.For noisy stage-consistent observations, the method interpolates measured frames and applies variance-adjusted features and filtered instruments.Source: §B.3 · p. 20

B.4 Gaussian polynomial and cross-noise correction

• Replacing Bernstein-basis monomials with variance-matched Q_k features removes Gaussian nonlinear-substitution bias in unnormalized moments.The correction holds when the supplied variance matches the true value.Source: §B.4 · p. 20
• Cross-pairing instruments and responses across independent noise-adjusted views removes their noise product in expectation.For correlated error covariance Σ and artificial variance τ^2I, the remaining cross-covariance is Σ − τ^2I; equal marginal variance alone does not ensure independent views.Source: §B.4 · p. 20
• The unbiasedness identities hold before row normalization and inversion, while normalization, constraints, regularization, and plug-in variance alter the finite-sample estimator.The completed inverse is analyzed with a conditional bound rather than covered by the moment-level unbiasedness result.Source: §B.4 · pp. 20–21

B.5 Proof of the conditional stability bound

• Testing the feasible minimizer's variational inequality with c⋆ and applying the triangle inequality yields Eq. (9).The proof sets A = K + Λ and defines the coefficient and prior-offset terms before applying the inequality.Source: §B.5 · p. 21
• The variational-inequality proof remains valid when positivity constraints are active.The feasible comparison vector is c⋆.Source: §B.5 · p. 21
• The bound is conditional on the realized finite moment system, not a uniqueness claim about an unrestricted physical law.Eq. (10) translates the coefficient bound through the evaluation-grid Gram matrix.Source: §B.5 · p. 21

C Architecture, training, and local sensitivity

• SCCT supplies a positive expert mixture and diagonal precision to a common constrained inverse rather than predicting the force directly.Physical moments stay at observation resolution; only neural context is averaged to 322.Source: C Architecture, training, and local… · p. 21
• The context uses 117 tokens from 13 frames and five inequality-constrained coefficients, with four attention heads dividing width 48 into 12 channels each.The small image and attention widths are fixed across moment-correction comparisons, holding network capacity constant.Source: C Architecture, training, and local… · p. 21
• Training proceeds from a degree-four initializer through mixed-degree and force-only continuations, but all three final SCCT seeds retain the force-only checkpoint.Thus the original checkpoint’s improvement cannot be attributed to degree supervision; labels, true coefficients, generating degree, and surface tension are unavailable to the map at inference.Source: C Architecture, training, and local… · p. 22
• With moments and precision fixed, mixing three constant experts varies the prior in an affine space of dimension at most two, and data-informed directions attenuate prior changes most strongly.With active bounds, the inverse relation applies to free coordinates.Source: C Architecture, training, and local… · p. 22
• At an interior solution, changing precision has a distinct local sensitivity from moving the prior; active-set changes instead follow the corresponding piece of the solution map.Together, the local relations distinguish prior movement from changes in its directional influence.Source: C Architecture, training, and local… · p. 22

D Correction and observation-source controls

• With the matching generator metadata, clean errors are at the numerical floor, whereas assuming an IMEX balance leaves a visible clean residual.The CNAB2 control uses an independently written force evaluation and its own discrete balance, separating time-scheme consistency from spatial observation error.Source: D Correction and observation-source… · pp. 22–23

E External weak-form baseline protocol

• The external weak-form comparison uses the same 24 ETDRK4 laws and noise arrays, but adds no Hermite correction, independent views, row normalization, or learned prior.Its EIV–IRLS row is a PDE adaptation of WENDy’s first-order covariance weighting, with a prescribed 48-test construction that has non-negligible clean error.Source: E External weak-form baseline protocol · p. 23, Table E.1 · p. 23

F Synthetic trajectories with specified physical parameters

• The three specified-parameter trajectories share one constitutive law and serve as demonstrations, not independent-law tests.Forecasts use the inferred force in the Fourier solver without learned field correction; exact-field restarts isolate constitutive error, while noisy observed restarts also contain initial-state error.Source: F Synthetic trajectories with… · pp. 23–24
• The inclusion-breakup example uses diffuse initial data formed by applying Q(d/ϵ) to signed distance, and its behavior includes finite-thickness diffuse-interface effects.Forecast evaluation includes relative field error, interface-band root-mean-square error, and positive-phase area error.Source: F Synthetic trajectories with… · p. 24

G Precision and structural controls

• The precision-only comparison holds one global expert mixture fixed while changing the precision rule, using six observation-only diagnostics and a shared inverse, coefficient floor, and precision bounds.Standardization uses training data only, and neural precision rules use three training seeds with validation checkpoint selection.Source: G Precision and structural controls · p. 24
• With precision fixed, the truth-informed simplex oracle leaves 0.0522 pp of mixing headroom, but the learned gate realizes only 0.00044 pp.The oracle uses unavailable true force and quantifies candidate complementarity, not learnability; the truth-dependent decomposition also associates SCCT improvement with a smaller prior term, with 16 favorable laws carrying most of the mean difference.Source: G Precision and structural controls · p. 24
• The planned normal-approximation power at a −0.005 pp alternative was about 94% for diagonal minus scalar and 6–13% for the other contrasts.These are design projections, not observed test power; the effect-size convention controls study sensitivity, not physical significance.Source: G Precision and structural controls · p. 24
• Across eight prespecified structural contrasts on distinct locked 96-law test sets, every multiplicity-adjusted interval includes zero.The experiments do not establish that extra expert or attention complexity improves the stronger baselines.Source: G Precision and structural controls · p. 24, Table G.2 · p. 25

H Information loss and departures from the noise and force models

• Underestimating independent Gaussian noise variance is more damaging than overestimating it, while spatially correlated Gaussian noise violates Eq. (15) and temporal correlation or Laplace noise biases temporal scale estimates.The controls delineate assumptions rather than extending exact correction beyond Gaussian independent errors.Source: H Information loss and departures from… · p. 25
• Combining raw moments from two or four independently observed initial fields reduces global error from 0.208% to 0.128% and 0.091% on the eight-law control.Late trajectories that have lost every interface have condition numbers between 1.2 × 10^9 and 10^16 and force errors above 22%; trajectories retaining an interface remain below 1.4 × 10^5 and 6.5%.Source: H Information loss and departures from… · p. 25
• Table H.1 reports observation-only detection of trajectories with force error above 1% on the 84-law family study, counting missed failures among 43 failures.The table caption identifies the detection scope and missed-failure denominator.Source: Table H.1 · p. 25
• Expert disagreement detects every tested failure but also alarms on some accurately recovered laws outside the training coefficient range; moment-equation residuals are less sensitive.The diagnostics’ false alarms and missed failures measure limits of model checking.Source: H Information loss and departures from… · p. 25

• Mean relative front-width errors are 0.017% for SCCT, 0.017% for the moment map, 0.016% for the global mixture, and 0.014% for the trajectory inverse.The width metric does not rank the identified methods in the same order as force or field error; noisy restarts also raise sharp/broad front field errors to approximately 0.035%/0.081%.Source: I Forecast, calibration uncertainty,… · p. 26
• Three-dimensional forecasts use fixed laws inferred from noisy planar trajectories, with no three-dimensional observations entering the inverse.The evaluation spans three unseen geometries and uses an interface-band error to avoid domination by the negative bulk phase.Source: I Forecast, calibration uncertainty,… · p. 26
• 0.848% interface-band error for a 3%-rescaled exact force contrasts with 0.0117% for SCCT and 0.0101% for the global inverse, despite the same sampled event.In the 128³ sensitivity screen, 10% and 30% rescalings retain the sampled event while errors rise to 2.79% and 8.04%.Source: I Forecast, calibration uncertainty,… · p. 27
• All four inference methods reproduce 107 of 108 event labels, while all tilted laws reproduce 108, so event agreement offers little method discrimination in this design.The one near-threshold coalescence distinguishes merging from vanishing; sensitivity ratios compare estimation errors with one imposed direction, not equal-norm force perturbations.Source: I Forecast, calibration uncertainty,… · p. 27
• A relative change in external surface tension changes the calibrated interface width linearly and the energy quadratically, without changing G or any forecast.Bootstrap uncertainty repeats the complete estimator after adding synthetic observation noise and, when specified, resampling surface tension.Source: I Forecast, calibration uncertainty,… · p. 27
• One 128², 61-frame evaluation takes approximately 0.118 s for the SCCT ensemble, 0.088 s for the moment map, and 2.65 s for the dense trajectory inverse.The global and smoothness inverses take approximately 0.086 s online and avoid neural training; timings are specific to the evaluated hardware.Source: I Forecast, calibration uncertainty,… · p. 27, Table I.2 · p. 28