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PhysDEM: Physics-Defined Energy-Matching Diffusion for Spatiotemporal Field Generation under Scarce Measurements

Zhenyu Liang, Yining Huang, Yubo Zhao, Jack C. P. Cheng

arXiv:2610.01759v1cs.CV

TL;DR

Scarce spatial measurements are insufficient to characterize distributions over complete fields, while conventional data-driven diffusion methods rely on full-field datasets. PhysDEM combines a physics-defined target with a learned energy-induced denoising correction and Gaussian-conditioned sampling; experiments report coherent recovery, efficient sampling, and stable diagnostics under tested noise levels.

  • Problem

    Conventional diffusion models for physical-field generation typically rely on complete-field datasets, which are unavailable when only scarce spatial measurements are present.

    Source: §1 · p. 1

  • Method

    PhysDEM reweights a measurement-conditioned Gaussian reference by PDE residual energy, trains an energy-determined conditional-mean denoiser, and uses a physics-displacement sampler with Gaussian conditioning for measurement changes without retraining.

    Source: §1 · p. 2

  • Results

    Experiments report coherent field recovery, efficient sampling, and stable diagnostics under tested noise levels.

    Source: §6 · p. 9

  • Takeaways & Limitations

    Site-informed cases illustrate real-world value for field assessment, while weakly informed static fields retain broad uncertainty.

    Source: §6 · p. 9

    The measurement setting assumes time-series observations at a handful of spatial locations with a linear observation model and additive Gaussian noise.

    Source: §3.2 · p. 3

Abstract

from arXiv · show

Generating and predicting spatiotemporal physical fields from scarce measurements is challenging, as observations are insufficient to characterize a distribution over complete fields. This limits conventional data-driven diffusion models that rely on full-field datasets. We introduce PhysDEM, a physics-defined diffusion framework that combines governing equations with spatially sparse observations to generate multiple plausible fields. First, we construct a Gibbs target by reweighting a measurement-conditioned Gaussian reference with PDE residual energy. Second, we derive an exact conditional-mean identity that reduces denoising to supervised learning of the standardized energy-induced mean correction. Third, a physics-displacement probability flow cancels Gaussian reference terms and enables amortized sampling with changing measurements through Gaussian conditioning, without retraining. Experiments on synthetic PDE systems and real-world-informed applications demonstrate that PhysDEM supports coherent field recovery and efficient sampling while maintaining stable diagnostics under tested noise levels, illustrating its practical value for field assessment. To our knowledge, PhysDEM is the first physics-defined diffusion model enabling amortized spatiotemporal field inference without preassembled full-field datasets.

1 INTRODUCTION

Existing physical-field diffusion methods generally depend on complete-field training data, which limits their use with only a few spatial measurements. PhysDEM instead defines a physics- and measurement-constrained target and uses it to train and reuse a sampler without preassembled full-field data.

1 INTRODUCTION

  • Existing diffusion approaches typically rely on complete-field datasets, while a handful of spatial measurements alone cannot characterize a distribution over complete fields.These approaches either learn physical patterns from large datasets or incorporate PDEs through training losses or sampling guidance.Source: §1 · p. 1
  • Table 1 compares methods across PDE use, training without full-field data, reuse across measurement counts, and full-field inference from sparse measurements.The caption defines these criteria but does not report comparative outcomes.Source: Table 1 · p. 2
  • PhysDEM defines a target using governing equations and measurements, enabling training on internally sampled complete fields without preassembled full-field datasets.The target is a Gibbs distribution formed by reweighting a Gaussian reference with PDE residual energy.Source: §1 · pp. 1–2
  • An exact conditional-mean identity makes denoiser training supervised learning of the energy-induced mean correction relative to an analytic Gaussian estimate.The denoiser implements energy matching against the physics-defined target.Source: §1 · p. 2
  • The physics-displacement probability-flow sampler cancels Gaussian reference terms and uses the learned normalized correction as velocity, reusing the model across measurement configurations without retraining.Changing measurements are accommodated through Gaussian conditioning.Source: §1 · p. 2

2 RELATED WORK

Prior physical-field generation methods use data-driven learning, PDE constraints, or energy-based sampling, but commonly retain data or reuse limitations. PhysDEM is presented as combining a PDE-defined target with sparse-measurement full-field inference and reuse across measurement counts.

2 RELATED WORK

  • Diffusion models offer non-adversarial denoising and flexible conditioning, while existing physical-field generators include GANs reliant on large radar archives, PI-GAN, and FNOPE requiring paired parameters and simulator outputs.PI-GAN can be susceptible to discriminator overfitting.Source: §2 · p. 2
  • PDE residual training losses and sampling guidance incorporate physical constraints, but these approaches generally rely on full-field training data from measurements or forward simulations.Learned-physics methods have not been demonstrated for full-field inference from scarce measurements alone.Source: §2 · pp. 2–3
  • Energy-based diffusion samplers can sample prescribed unnormalized densities without precomputed target datasets, but their original formulations do not explicitly support reuse across measurement counts without retraining.The paper contrasts this with PhysDEM's PDE-based self-sampled training target and reuse across measurement counts.Source: §2 · p. 3

3 PRELIMINARIES

The preliminaries represent the physical system through a joint static-parameter and time-varying state field, with PDE residuals measuring discrete equation discrepancies. Measurements are modeled as sparse, noisy linear observations that support Gaussian conditioning.

3.1 SPATIOTEMPORAL PHYSICAL FIELDS AND PDE RESIDUALS

  • The physical system is a possibly vector-valued state field over space and time governed by a first-order evolution equation with a static parameter field.The complete candidate field jointly contains discretized static parameters and dynamic states, whose physical coupling is encoded through PDE residual energy.Source: §3.1 · p. 3
  • Discrete PDE residuals measure local discrepancies at interior grid points, vanish when the corresponding equations are satisfied, and incorporate boundary conditions through the discrete operators.These residuals form the basis of the PDE residual energy.Source: §3.1 · p. 3

3.2 SCARCE MEASUREMENTS

  • Measurements are time series from a handful of spatial locations, modeled as linear observations with additive Gaussian noise.Measurement operators extract or average field values; positive-definite noise covariances and independence assumptions support closed-form Gaussian conditioning.Source: §3.2 · p. 3

4 METHODOLOGY

PhysDEM defines a measurement-constrained Gibbs target over complete fields, trains a denoiser from an exact conditional-mean identity, and uses a physics-displacement flow for amortized sampling. Measurement configurations update the Gaussian reference while the PDE residual energy and trained model remain fixed.

4.1 PHYSICS-DEFINED AND MEASUREMENT-CONSTRAINED TARGET DISTRIBUTION

  • The target distribution reweights a measurement-conditioned Gaussian reference by PDE residual energy, combining measurement constraints with physical consistency.The framework has three stages: target definition, denoiser training, and amortized sampling.Source: §4.1 · p. 3, §4 · p. 3

4.1.1 MEASUREMENT-CONDITIONED GAUSSIAN REFERENCE

  • Conditioning the Gaussian base on a measurement configuration yields a reference whose covariance correlations allow measurements to inform unmeasured locations.The base covariance describes spatial correlations in z and spatiotemporal correlations in u; the Gaussian reference incorporates measurements but does not itself enforce the PDEs.Source: §4.1.1 · pp. 3–4, §4.1.2 · p. 4

4.1.2 PHYSICS-TILTED GIBBS DISTRIBUTION

  • The PDE residual energy reweights the Gaussian reference, with a quadratic linearization absorbed into an augmented reference and the remainder held fixed across measurement configurations.The energy weights fields with larger residual energy less; the linearization uses a global expansion point fitted without the measurement likelihood.Source: §4.1.2 · pp. 4–5

4.2 ENERGY-DETERMINED CONDITIONAL-MEAN DIFFUSION DENOISER

  • For a perturbed target field, the energy-weighted Gaussian mean equals the conditional mean of the original field, providing the squared-error-optimal denoising estimate.The Gaussian conditional mean is the analytic estimate; the PDE residual energy determines its correction.Source: §4.2 · p. 5
  • Training predicts the standardized energy-induced mean correction from Gaussian estimates, coordinate standard deviations, and the signal coefficient, using sampled fields as correction labels.This avoids evaluating the expectations defining the energy-weighted mean directly; the covariance enters the network only through its diagonal.Source: §4.2 · p. 6

4.3 PHYSICS-DISPLACEMENT PROBABILITY-FLOW SAMPLER

  • The probability-flow ODE cancels Gaussian-reference terms, so the learned normalized correction supplies the sampling velocity and supports reuse with updated measurement-conditioned inputs without retraining.Trajectories update the full field jointly across space and physical time, using Heun steps up to amax < 1.Source: §4.3 · p. 6

5 EXPERIMENTS AND RESULTS

Across the tested cases, PhysDEM reuses a fixed network across measurement counts, samples efficiently, and maintains diagnostics near reference scales under tested noise levels. Ablations show trade-offs among the reported diagnostics.

5 EXPERIMENTS AND RESULTS

  • PhysDEM is evaluated on three synthetic PDEs—Darcy, advection–diffusion, Fisher–KPP—and two real-world-informed applications, SPE10 and Bemidji, comparing scarce-measurement field recovery with baselines and examining reuse, ablations, sampling cost, and noise sensitivity.Source: §5 · p. 7

Reuse across measurements

  • 10.3–23.5% decreases in full-field mean pointwise 90% interval widths occur from one to eight wells across all five cases, with one fixed network per case.Observation and residual diagnostics remain near their reference scales across K = 1, 2, 4, 8.Source: §5 · p. 8

Ablation study

  • Without reference augmentation or with τ-scaled labels, the largest mean-rank discrepancy rises from 2.18 to 11.65 or 11.84, while learned velocity raises SSRg from 0.54 to 0.89.Tolerance 0.15 moves SSRg closer to one (1.08), but fails the rank check and roughly doubles the residual ratio.Source: Table 3 · p. 8, §5 · p. 9

Sampling time and noise sensitivity

  • 1.47 s/member on Darcy makes PhysDEM faster than six of eight baselines, including DiffusionPDE (4.38 s/member), while observation misfit is closest to one.Across five cases, observation and residual diagnostics remain near one for noise variance multipliers 0.1, 1, and 10.Source: §5 · p. 9

6 CONCLUSION AND LIMITATIONS

PhysDEM supports coherent field recovery and efficient sampling without external full-field datasets, but weakly informed static fields retain broad uncertainty; the authors propose richer measurements for future recovery.

6 CONCLUSION AND LIMITATIONS

  • Weakly informed static fields retain broad uncertainty, and future work will use richer measurements to improve joint static and dynamic field recovery.The conclusion also reports coherent recovery, efficient sampling, and stable diagnostics under tested noise levels.Source: §6 · p. 9

A SUPPORTING PRELIMINARIES

  • The appendices state conventions and assumptions and derive Gaussian identities underlying target construction, denoiser training, and sampling.Source: A SUPPORTING PRELIMINARIES · p. 12

A.1 NOTATION AND MATHEMATICAL TOOLS

  • The notation treats static parameter and state fields jointly as a finite-dimensional field, with physical time a coordinate of the state and measurements grouped by observation setup and values.Source: §A.1 · p. 12

A.2 ASSUMPTIONS

  • The setup assumes a linear observation model with additive Gaussian noise in the modeled coordinates and holds calibrated target parameters fixed as measurement configurations change.The stated assumptions also require finite normalizers and moments for the relevant Gaussian queries.Source: §A.2 · pp. 12–13

A.3 GAUSSIAN IDENTITIES FOR QUADRATIC ENERGIES

  • Quadratic PDE energies can be absorbed analytically into a Gaussian reference, providing an augmented reference for denoising and local references for training chains.For positive penalty strength, the tilt is equivalent to conditioning on a virtual linear observation representing the physical penalty.Source: §A.3 · p. 13

B TARGET DEFINITION, CALIBRATION, AND GAUSSIAN AUGMENTATION

  • The Gaussian reference specifies field variability and measurement information, while PDE residual energy reweights it; Gaussian augmentation exposes an analytic part of the physics without changing the target.Source: B TARGET DEFINITION, CALIBRATION, AND… · p. 13

B.1.1 COVARIANCE SPECIFICATION

These sections specify the Gaussian reference and measurement conditioning, define a selected-residual Gibbs target, and describe its calibration and Gaussian augmentation.

B.1.1 COVARIANCE SPECIFICATION

  • The base Gaussian covariance encodes assumed spatial and spatiotemporal structure, while PDE residual energy couples the otherwise independent static and state fields.Source: §B.1.1 · pp. 13–14

B.1.2 MEASUREMENT CONDITIONING AND MATHERON’S RULE

  • Gaussian conditioning propagates measurement information to unmeasured locations, and Matheron’s rule samples the conditional reference without factoring its full covariance.The conditioning solve is in measurement space, whose dimension is the number of scalar measurements.Source: §B.1.2 · p. 14

B.1.3 FFT-BASED GAUSSIAN OPERATIONS

  • For stationary covariance blocks, circulant embedding enables FFT-based covariance products and Gaussian draws at O(M log M) cost per block.An overall O(d log d) cost requires bounded embedding overhead, and spectral clipping constitutes a covariance approximation.Source: §B.1.3 · p. 14

B.2.1 RESIDUAL ENERGY AND INFORMATION PROJECTION

  • At finite penalty strength, selected PDE and initial-value residuals enter as soft penalties, and the Gibbs target changes their relative weights against the Gaussian reference.The residual score defines the physical reweighting of the reference.Source: §B.2.1 · pp. 14–15
  • For an expected residual-score constraint, the Gibbs target uniquely minimizes relative entropy from the Gaussian reference among feasible distributions.A unique positive penalty multiplier gives the specified score when it lies between the reference expectation and the essential infimum.Source: §B.2.1 · p. 15

B.2.2 FOURIER REPRESENTATION OF THE PDE RESIDUALS

  • The residual representation retains selected space–time Fourier components, scales them to balance reference amplitudes, and combines them with initial-value residuals using specified weights.Calibration applies only to this selected representation and does not itself bound unselected residual components.Source: §B.2.2 · p. 15

B.2.3 PENALTY INITIALIZATION AND REFINEMENT

  • The penalty initializer substitutes the retained spectral row count for effective rank, so it initializes calibration rather than giving an exact full-target tolerance identity.Finite-chain estimates and response approximation likewise make the tolerance calibration approximate rather than an exact moment-matching guarantee.Source: §B.2.3 · p. 16

B.2.4 STATIC MARGINALS AND SCALAR MEAN CALIBRATION

  • Scalar mean calibration adjusts the base mean while preserving its covariance, but it targets one moment rather than the full static marginal and later measurements may shift that mean.Finite-chain moment estimates make the numerical adjustment approximate.Source: §B.2.4 · pp. 16–17

C TRAINING THE ENERGY-DETERMINED CONDITIONAL-MEAN DIFFUSION

The appendix derives the conditional-mean denoiser and regression labels, then shows how Gaussian perturbations preserve the reference and connect it to the Gibbs target.

C TRAINING THE ENERGY-DETERMINED CONDITIONAL-MEAN DIFFUSION

  • The appendix derives the conditional-mean denoiser, its energy-induced correction, and target-sample regression labels under a fixed calibrated model.Its denoising queries require finite normalizers and polynomial moments; other Gaussian queries need their own integrability conditions.Source: C TRAINING THE ENERGY-DETERMINED… · pp. 17–18

C.1 REFERENCE PRESERVATION

  • The reference-preserving perturbation leaves the augmented Gaussian marginal unchanged while varying the signal coefficient from independent noise to the original field.At coefficient zero the perturbed field is independent of the original; at one it equals the original field.Source: §C.1 · p. 18
  • Under the Gibbs target, the same perturbation connects the Gaussian reference at signal coefficient zero to the target at one, and the denoiser correction accounts for this change.Source: §C.1 · p. 18

C.2 PROOF OF THEOREM 1

The appendix establishes the energy-tilted conditional-mean identity and develops its role in regression training and probability-flow sampling, including conditions for preserving the intended correction and target. It also addresses fixed-model conditioning with changing measurements, their effect on field uncertainty, and credible-set coverage in light of model and sampler discrepancies.

C.2 PROOF OF THEOREM 1

  • The conditional target mean is D(m_a(x_a), Σ_a); with regularity, its correction is a covariance-scaled conditional average of the remaining energy gradient.The quadratic energy contribution is represented analytically, and the small-noise correction converges to −τ C̃∇E(m) under concentration and gradient-integrability conditions.Source: §C.2 · p. 19

C.3 TARGET DRAWS AS UNBIASED REGRESSION LABELS

  • Target draws provide supervised labels for the standardized energy-induced correction without evaluating energy-weighted or energy-gradient expectations.Standardization accounts for coordinate-dependent uncertainty but does not decorrelate labels; in the quadratic case each standardized coordinate variance is at most one, while finite-chain training draws can only approximate the exact target population.Source: §C.3 · p. 20

C.4 LOCAL GAUSSIAN REFERENCES FOR TRAINING; C.5 PCN SAMPLING OF THE GIBBS TARGET

  • Local quadratic Gaussian references define pCN proposals, while the acceptance correction retains the full residual energy and the fixed-kernel chain satisfies detailed balance for π_C.Finite-chain output remains approximate: freezing the kernel does not establish equilibration, and thinning does not guarantee independence.Source: §C.4 · pp. 20–21, §C.5 · p. 21

C.6 COVARIANCE ENCODING AND THE REGRESSION TARGET

  • The population predictor is the conditional projection of the standardized correction onto encoded inputs; it is exact when those inputs determine the covariance, but diagonal encoding can leave correlation-dependent error.For shared known correlation K, coordinate scales recover Σ_a and the encoding-error term vanishes; reuse on new measurements still depends on encoding adequacy and generalization.Source: §C.6 · p. 22

C.7 TRAINING CONFIGURATIONS

  • Training configurations and complete fields are generated under the fixed physical model, so training needs neither an external preassembled full-field dataset nor measurements later supplied at inference.Simulated observations and approximate conditional target samples produce perturbed fields and regression labels for training.Source: §C.7 · p. 22

D SAMPLING WITH THE PHYSICS-DISPLACEMENT PROBABILITY FLOW; D.1 PROBABILITY FLOW AND NUMERICAL SAMPLING

  • The probability-flow sampler cancels Gaussian-reference score terms, leaving the learned energy correction as its velocity; under stated regularity conditions, the exact flow converges from the Gaussian reference to π_C.The implemented Heun sampler stops at a_max < 1 without a terminal correction; for affine residuals, Gaussian conditioning already produces target draws and the flow velocity is zero.Source: §D.1 · pp. 23–24

D.2 CONDITIONING A FIXED MODEL

  • Adding measurements by Gaussian conditioning updates the reference while leaving the remaining energy and exact denoiser map unchanged.Conditioning the augmented reference is equivalent to conditioning the base on all measurements and then applying the same fixed augmentation; each observation is incorporated once.Source: §D.2 · p. 25
  • A fixed trained network supports new measurement layouts, values, and counts by rebuilding the Gaussian reference, separating training cost from per-query conditioning and learned-flow computations.The network is reused with the resulting Gaussian inputs, without retraining.Source: §D.2 · p. 25

D.3 FIELD VARIABILITY UNDER ADDITIONAL MEASUREMENTS

  • Additional measurements reduce reference uncertainty in directions correlated with observed quantities, including unmeasured field values; nonlinear energies imply an average covariance reduction.A direction is unchanged precisely when A ˜Cw = 0; under the fixed model, expected conditional covariance is no larger than the original in positive-semidefinite order. For affine residuals, the exact target is Gaussian, and the covariance ordering holds for every measurement value.Source: §D.3 · p. 25

D.4 CREDIBLE SETS AND EXPECTED COVERAGE

  • Generated ensembles represent uncertainty in quantities derived from complete fields, while the coverage result separates the model’s contribution from sampler and credible-set construction accuracy.Source: §D.4 · p. 25

D.4.1 CONDITIONAL LAWS AND COMPUTATIONAL DISCREPANCIES

  • Under a shared likelihood and finite projected joint KL, the analysis distinguishes true, exact-model, and implemented-sampler conditional laws and assesses coverage using sampler TV discrepancy and set-probability deficit.The set probability is evaluated on a fresh draw independent of the randomness used to construct the set, and both terms average over the true measurement marginal.Source: §D.4.1 · p. 26

D.4.2 EXPECTED COVERAGE BOUND

The expected coverage bound relates reported-set coverage to model discrepancy, sampling approximation, and set construction under a fixed model and observation likelihood. Under correct specification and exact sampling, expected true coverage equals the expected set probability; with approximation, the bound quantifies a lower bound.

D.4.2 EXPECTED COVERAGE BOUND

  • Theorem 3 requires finite projected discrepancy Dg, not finite field-level KL.When field-level KL is finite, the projected discrepancy is bounded by KL(π*∥πC0).Source: §D.4.2 · p. 27
  • Under correct specification, exact sampling, and expected set probability α, expected true coverage equals α; with approximation, equation 91 gives a lower bound.The result averages over true fields, measurements, and set construction conditional on H; it does not guarantee coverage for every measurement outcome.Source: §D.4.2 · p. 27
  • The bound has explicit contributions from model discrepancy, sampling approximation, and set construction for a fixed model and observation likelihood.The theorem's probability averages over the true field, measurements, and set construction, conditional on H.Source: §D.4.2 · p. 27

E.1 CASES AND DATA SOURCES

The cases span synthetic PDE systems, a heterogeneous reservoir benchmark, and a Bemidji-informed transport setting. They vary in field, measurement, and data construction, with the Bemidji comparison explicitly descriptive rather than independent held-out validation.

Synthetic cases

  • The synthetic cases use 64×64 spatial grids, 48 time states, 64-member ensembles, and noisy measurements; the baseline comparison uses K = 4 wells.Measurement reuse tests K ∈{1, 2, 4, 8} with a fixed checkpoint per case; synthetic reference fields are sampled by pCN.Source: §E.1 · p. 27

Darcy flow

  • Darcy flow models static log-permeability and pressure with no-flow boundaries and zero initial pressure.The prescribed injection–extraction source uses storage coefficient cD = 1.Source: §E.1 · p. 27

Advection–diffusion

  • The advection–diffusion case models static streamfunction, source, and concentration, with diffusivity Dc = 2 and decay rate κ = 0.5.It starts from zero concentration and uses zero exterior ghost-cell concentrations for advective and diffusive boundary fluxes.Source: §E.1 · p. 27

Fisher–KPP

  • Fisher–KPP models static log-diffusivity and state with growth rate r = 2; Darcy and Fisher–KPP observe both unknown fields, while advection–diffusion observes concentration only.The Fisher–KPP initial field comprises three Gaussian seeds clipped at one; the cases use cell-centered finite volumes and specified time-stepping schemes.Source: §E.1 · p. 27

SPE10

  • The SPE10 case uses a resampled, centered, and 0.5-scaled window of Model 2, layer 10 permeability structure to inform the static prior scales.The pressure reference is obtained by solving the Darcy equation on a finer grid.Source: §E.1 · p. 27

Reservoir benchmark

  • The reservoir assessment block-averages a 256×256 grid onto the inference grid and adds Gaussian noise to log-permeability and pressure measurements at selected wells.This is described as a controlled field-recovery assessment from scarce measurements in a heterogeneous reservoir benchmark.Source: §E.1 · p. 28

Bemidji

  • Bemidji uses 147 benzene records from 38 wells across seven campaigns, but its 56 conditioning values are synthetic noisy measurements at real well locations and times.Comparison with the 147 records assesses agreement with source data and is descriptive, not independent held-out validation.Source: §E.1 · p. 28

E.2 BASELINE IMPLEMENTATIONS AND EVALUATION

The evaluation describes task-adapted baseline implementations and diagnostic conventions, then examines measurement reuse across well counts. Reuse narrows some full-field intervals, but static-field uncertainty and calibration behavior vary by case.

E.2 BASELINE IMPLEMENTATIONS AND EVALUATION

  • The eight task-adapted baselines use 4,096 training and 512 validation fields sampled approximately from the fixed physics-defined target, whereas PhysDEM uses internal approximate pCN pools.The archived FNOPE implementation generates static parameters and completes states with a deterministic forward solve.Source: §E.2 · p. 28
  • Residual scale one is a prescribed reference, not a universal optimum, and deterministic state completion with zero residual does not establish joint posterior calibration.The mean-rank checks are not full rank-uniformity tests, and comparisons are descriptive because truth pools and rank conventions differ across archives.Source: §E.2 · pp. 28–29
  • Darcy Figures 2 and 7 show saved samples, static-field spread, pressures, and observation residuals; hit counts describe one well and reference, not simultaneous trajectory coverage.Bemidji’s plotted latent-field intervals contain 56 of 147 source records, which also inform reference construction.Source: §E.2 · pp. 29–30, Figure 7 · p. 29

E.3 REUSE ACROSS MEASUREMENT COUNTS

  • Each case reuses one checkpoint and reference field for K ∈ {1, 2, 4, 8}, with 64 samples per configuration; dynamic observation noise is regenerated across K.The configurations are not sequential additions retaining all previous observation values; observation counts are 49K, 48K, or 7K scalar values depending on case.Source: §E.3 · p. 30
  • SPE10’s full-field interval width decreases 11.2% from K = 1 to K = 8, while static-field standard deviation rises from 0.660 to 0.677 and CK falls from 0.851 to 0.687.The results caution that interval contraction does not imply improved calibration or shrinking uncertainty in every field component.Source: Figure 8 · p. 30, §E.3 · p. 31
  • For concentration-only advection–diffusion and Bemidji, ψ relative L2 errors at K = 4 are 1.028 and 1.019, while spatial RMS spread changes by less than 0.2% from K = 1 to K = 8.These results are consistent with weak constraints on ψ, but do not establish that its posterior equals its prior.Source: §E.3 · p. 31

E.4 COMPONENT ABLATIONS AND TOLERANCE SENSITIVITY

The ablations and sensitivity analyses show trade-offs among calibration diagnostics, tolerance choices, and sampling cost. Results support stability under tested noise-variance changes, while the archived timing comparison is not a controlled end-to-end benchmark.

E.4 COMPONENT ABLATIONS AND TOLERANCE SENSITIVITY

  • The no-augmentation ablation also reflects a training–inference reference mismatch, while the reference-only variant’s SBC entry was not evaluated.Evaluation scripts still construct augmented references even when the training augmentation flag is disabled.Source: §E.4 · p. 31
  • A spread–error ratio closer to one does not establish improvement in every diagnostic.The 0.15 tolerance’s closer-to-one SSRg coincides with a mean-rank statistic of 5.20.Source: §E.4 · p. 31
  • SSRg = 1.08 at tolerance 0.15 is closer to one than the full model’s 0.89, but its mean-rank statistic is 5.20.Ratios using the common main tolerance are 0.35 and 2.10 for the 0.05 and 0.15 variants; their own squared tolerances yield approximately 1.12 and 0.76.Source: §E.4 · p. 31

E.5 SAMPLING COST AND NOISE SENSITIVITY

  • 1.47 s/member for PhysDEM versus 4.38 s/member for DiffusionPDE is an archived timing comparison, not a controlled end-to-end hardware benchmark.PhysDEM timing includes sampling, summary-statistic computation, and CPU transfer, but excludes reference construction, residual scoring, and offline training.Source: §E.5 · pp. 31–32
  • Across 15 noise-variance settings, zobs ranges from 0.906 to 1.146 and rPDE from 0.974 to 1.120, supporting stability of reported diagnostics under tested noise changes.The test changes the covariance by factors of 0.1, 1, or 10; it does not test misspecified noise or establish full posterior calibration.Source: §E.5 · p. 32

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