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

arXiv: https://arxiv.org/abs/2607.01749 · HTML: https://mlsanity.com/paper/2607.01749/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 18 of its 65 references that are in the map.
- First posted on arXiv 2 Jul 2026.

## Builds on (most cited first, 18 of 18)

- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Scientific Machine Learning through Physics-Informed Neural Networks: Where we are and What's next](https://mlsanity.com/paper/2201.05624/) (2022)
- [On the eigenvector bias of Fourier feature networks: From regression to solving multi-scale PDEs with physics-informed neural networks](https://mlsanity.com/paper/2012.10047/) (2020)
- [An Expert's Guide to Training Physics-informed Neural Networks](https://mlsanity.com/paper/2308.08468/) (2023)
- [Encoding physics to learn reaction-diffusion processes](https://mlsanity.com/paper/2106.04781/) (2021)
- [Solving Inverse Stochastic Problems from Discrete Particle Observations Using the Fokker-Planck Equation and Physics-informed Neural Networks](https://mlsanity.com/paper/2008.10653/) (2020)
- [Systems Biology: Identifiability analysis and parameter identification via systems-biology informed neural networks](https://mlsanity.com/paper/2202.01723/) (2022)
- [A cusp-capturing PINN for elliptic interface problems](https://mlsanity.com/paper/2210.08424/) (2022)
- [Identifying Drift, Diffusion, and Causal Structure from Temporal Snapshots](https://mlsanity.com/paper/2410.22729/) (2024)
- [BiLO: Bilevel Local Operator Learning for PDE Inverse Problems](https://mlsanity.com/paper/2404.17789/) (2024)
- [Inferring stochastic dynamics with growth from cross-sectional data](https://mlsanity.com/paper/2505.13197/) (2025)
- [Position: Biology is the Challenge Physics-Informed ML Needs to Evolve](https://mlsanity.com/paper/2510.25368/) (2025)
- [Cox process representation and inference for stochastic reaction-diffusion processes](https://mlsanity.com/paper/1601.01972/) (2016)
- [Adaptive Collocation Point Strategies For Physics Informed Neural Networks via the QR Discrete Empirical Interpolation Method](https://mlsanity.com/paper/2501.07700/) (2025)
- [Bayesian BiLO: Bilevel Local Operator Learning for Efficient Uncertainty Quantification of Bayesian PDE Inverse Problems with Low-Rank Adaptation](https://mlsanity.com/paper/2507.17019/) (2025)
- [Numerical PDE solvers outperform neural PDE solvers](https://mlsanity.com/paper/2507.21269/) (2025)
- [Physics-Informed Neural Networks for Biological 2D+t Reaction-Diffusion Systems](https://mlsanity.com/paper/2604.18548/) (2026)
- [Using neural networks to estimate parameters in spatial point process models](https://mlsanity.com/paper/2109.15056/) (2021)

## Abstract (arXiv)

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.