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arxiv: 2607.01749 · v1 · pith:UOTJEKNJnew · submitted 2026-07-02 · 🧬 q-bio.QM · physics.bio-ph· stat.ML

Identifiability Limits of Physics-Informed Inference for Spatial Stochastic Dynamics from Static Snapshots

Rujie Gu , Ray Zirui Zhang , Christopher E. Miles This is my paper

Pith reviewed 2026-07-03 02:27 UTC · model grok-4.3

classification 🧬 q-bio.QM physics.bio-phstat.ML
keywords structural identifiabilityphysics-informed machine learningspatial stochastic dynamicsstatic snapshotspoint sourcedistributed sourcegene expressioninference limits
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The pith

Distributed sources are non-identifiable from static snapshots, but a point source restores identifiability.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper asks when a static spatial pattern of molecules can identify spatially varying diffusivity, creation, destruction, and boundary exchange. A structural identifiability analysis shows that distributed sources are non-identifiable, whereas a point source such as a transcription site can restore identifiability. These limits are shaped by boundary conditions, spatial regularity of the dynamics, and the stochastic calculus convention. Physics-informed schemes can recover spatial heterogeneities from a single snapshot when the conditions are met.

Core claim

A structural identifiability analysis shows that distributed sources are non-identifiable, whereas a point source such as a transcription site can restore identifiability. These limits are further shaped by boundary conditions, the spatial regularity of the underlying dynamics, and even the stochastic calculus convention. Adapted physics-informed schemes demonstrate effective inference from a single snapshot.

What carries the argument

Structural identifiability analysis for physics-informed inference of spatial stochastic dynamics from static snapshots.

If this is right

  • Physics-informed inference can recover spatial heterogeneities from static data if a point source is present.
  • Modeling choices like boundary conditions and stochastic calculus affect what can be identified.
  • Careful identifiability analysis is required for meaningful interpretation of results from such inference.
  • Effective inference is possible from a single snapshot under identifiable conditions.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The findings suggest that in gene expression studies, the presence of transcription sites is crucial for inferring dynamics from snapshots.
  • Similar identifiability limits may apply to other spatial biological processes without localized sources.
  • Adding more data types could potentially overcome the non-identifiability of distributed sources.

Load-bearing premise

The identifiability results depend on specific modeling choices including boundary conditions, spatial regularity of the dynamics, and the stochastic calculus convention used.

What would settle it

Numerical experiments showing unique recovery of distributed source parameters or a change in identifiability when switching the stochastic calculus convention would test the central claims.

Figures

Figures reproduced from arXiv: 2607.01749 by Christopher E. Miles, Ray Zirui Zhang, Rujie Gu.

Figure 1
Figure 1. Figure 1: Schematic of the inverse problem. Particles are produced by a source B(x), move with spatially varying diffusivity D(x), and degrade. At steady state they are observed only through a static snapshot of their positions, which form a spatial Poisson process with intensity u(x), random in both number and location. The inverse problem is to recover features of the underlying dynamics, such as D(x) and the sour… view at source ↗
Figure 2
Figure 2. Figure 2: Same-density diffuse-source alternative under Itô dynamics with Dirichlet boundaries. Here γ = 10, D1(x) = 1.5 + sin(4x), and b1(x) = 1 σ √ 2π exp − (x − 0.5)2 /2σ 2  with σ = 0.1; the source perturbation is b2 = b1 + 1. The compensating D2 = D1 + δD follows the construction in Theorem 3.1. Panels show (a) the common density, (b) the indistinguishable diffusivities D1 and D2, and (c) the alternative sourc… view at source ↗
Figure 3
Figure 3. Figure 3: Same-density Itô alternative under absorbing Dirichlet boundaries. Here D1(x) = 2 + 0.8 sin(2πx), z = 0.5, γ = 5, b (1) 0 = 10, and b (2) 0 = 14; the alternative D2 = D1 + δD follows the construction in Theorem 4.6. Panels show (a) the common density, (b) the indistinguishable diffusivities D1 and D2, with δD continuous but kinked at z, and (c) the alternative source strengths. 0 0.2 0.4 0.6 0.8 1 x 1.4 1.… view at source ↗
Figure 4
Figure 4. Figure 4: Same-density Itô alternative under reflecting (Neumann) boundaries. Here D1(x) = 2 + 0.8 sin(2πx), b0 = 10, z = 0.5, and γ = 5; the alternative D2 = D1 + δD uses the homogeneous mode C = 1.5 from Theorem 4.6. Panels show (a) the common density, (b) the indistinguishable diffusivities D1 and D2, with δD continuous but kinked at z, and (c) the unchanged source strength. Theorem 4.7 (Fickian point-source iden… view at source ↗
Figure 5
Figure 5. Figure 5: Same-density Fickian alternative under absorbing Dirichlet boundaries. Here D1(x) = 2 + 0.5 cos(4πx), z = 0.5, γ = 5, b (1) 0 = 10, and b (2) 0 = 14; the alternative D2 = D1 + δD follows the construction in Theorem 4.7. Panels show (a) the common density, with the source kink at z, (b) the indistinguishable diffusivities D1 and D2, with δD continuous but kinked at z, and (c) the alternative source strength… view at source ↗
Figure 6
Figure 6. Figure 6: Recovery under Itô dynamics with Dirichlet boundaries, for Dtrue(x) = 0.10 + 0.05 sin(8πx) at source strengths b0 = 250 (a, b) and b0 = 1000 (c, d). In panels (a, c) the dotted black curve is the true D(x), the colored curves are the per-method median recovered D(x), and the shaded bands span the 10th to 90th percentile over 100 trials. In panels (b, d) the bars are the distribution of the inferred source … view at source ↗
Figure 7
Figure 7. Figure 7: Recovery under Fickian dynamics with Neumann boundaries, for the high-frequency target Dtrue(x) = 0.10 + 0.05 sin(8πx) at b0 = 1000. Even at this source strength only DTO tracks the oscillations of D(x), while PINN and BiLO smooth them out. Plotting conventions follow [PITH_FULL_IMAGE:figures/full_fig_p014_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Recovery under Fickian dynamics with Neumann boundaries for the lower-frequency target Dtrue(x) = 0.10 + 0.05 sin(6πx) at b0 = 250 (a, b) and b0 = 1000 (c, d). All three methods recover the oscillations, and the reconstruction tightens as b0 increases. Plotting conventions follow [PITH_FULL_IMAGE:figures/full_fig_p015_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Same-density alternative for Itô diffusion with unknown Robin permeabilities and fixed source strength. Here D1(x) = 2 + 0.5 sin(2πx), b0 = 10, z = 0.5, γ = 5, and κ0 = κL = 2; the perturbation uses flux offset c = 0.6 and follows the construction in Theorem 6.1. Panels show (a) the common density, (b) the indistinguishable diffusivities D1 and D2, (c) the unchanged point source, and (d) the alternative en… view at source ↗
read the original abstract

Despite increasing scale and resolution, many biological measurements remain destructive, revealing only spatial information rather than the dynamics it encodes. By combining flexible representations with mechanistic constraints, physics-informed machine learning offers a promising route to inferring these dynamics from static snapshots. Motivated by subcellular imaging of gene expression, we ask when a static spatial pattern of molecules can identify spatially varying diffusivity, creation, destruction, and boundary exchange, and how different inference schemes perform on the task. A structural identifiability analysis shows that distributed sources are non-identifiable, whereas a point source such as a transcription site can restore identifiability. These limits are further shaped by seemingly innocuous modeling choices: the boundary conditions, the spatial regularity of the underlying dynamics, and even the stochastic calculus convention. We then adapt several physics-informed schemes, differing in how they represent the solution and enforce the governing equations, and demonstrate effective inference from a single snapshot. Physics-informed approaches can thus recover spatial heterogeneities of biological dynamics from static data, but their use should be accompanied and guided by careful identifiability analysis for meaningful interpretation of the results.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The manuscript conducts a structural identifiability analysis of physics-informed inference for spatially varying parameters (diffusivity, creation, destruction, boundary exchange) in stochastic dynamics models, using only static spatial snapshots as data. It establishes that distributed sources are non-identifiable while point sources (e.g., transcription sites) can restore identifiability; these limits depend explicitly on boundary conditions, spatial regularity of the dynamics, and the stochastic calculus convention. The authors then adapt multiple physics-informed schemes that differ in solution representation and equation enforcement, and demonstrate that they recover spatial heterogeneities from a single snapshot in the motivating application of subcellular gene-expression imaging.

Significance. If the identifiability results and inference demonstrations hold, the work supplies a needed cautionary framework for physics-informed machine learning in quantitative biology, where destructive measurements often yield only static spatial data. Explicitly mapping how seemingly innocuous modeling choices (boundary conditions, regularity, Itô vs. Stratonovich) alter identifiability is a concrete strength that can guide model selection. The successful single-snapshot inference examples under the identified conditions further increase practical utility for subcellular imaging studies.

minor comments (3)
  1. The abstract states that identifiability limits depend on the stochastic calculus convention, but the main text should include an explicit side-by-side comparison (e.g., in a dedicated subsection or table) of the two conventions applied to the same model to make the dependence transparent to readers.
  2. When the different physics-informed schemes are introduced, a brief tabular summary of how each represents the solution and enforces the governing equations would improve readability and allow direct comparison of their performance on the identifiability-restoring point-source case.
  3. Figure captions should explicitly note which boundary condition and stochastic calculus convention were used for each panel so that readers can immediately connect the visuals to the identifiability claims.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary and significance assessment of the manuscript, as well as the recommendation for minor revision. We appreciate the recognition that the identifiability results and single-snapshot inference demonstrations provide a useful cautionary framework for physics-informed methods in quantitative biology.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper performs a structural identifiability analysis on a physics-informed inference setup for spatial stochastic dynamics, explicitly stating dependence on boundary conditions, spatial regularity, and stochastic calculus convention. No load-bearing steps reduce by construction to fitted parameters, self-citations, or renamed inputs; the central claims about non-identifiability of distributed sources and restoration by point sources follow from the stated modeling choices without self-referential closure. The derivation remains self-contained against external mathematical benchmarks for identifiability.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

The abstract provides no details on free parameters, axioms, or invented entities used in the analysis.

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