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arxiv: 2604.18548 · v1 · submitted 2026-04-20 · 💻 cs.LG · q-bio.QM

Recognition: unknown

Physics-Informed Neural Networks for Biological 2D{+}t Reaction-Diffusion Systems

William Lavery , Jodie A. Cochrane , Christian Olesen , Dagim S. Tadele , John T. Nardini , Sara Hamis
Authors on Pith no claims yet

Pith reviewed 2026-05-10 05:14 UTC · model grok-4.3

classification 💻 cs.LG q-bio.QM
keywords physics-informed neural networksbiologically-informed neural networksreaction-diffusion systemsequation discoverycell population dynamicstime-lapse microscopysymbolic regression
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The pith

A neural framework learns closed-form reaction-diffusion equations for 2D+t biological systems like lung cancer cell growth from microscopy data.

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

The paper extends biologically-informed neural networks to two spatial dimensions plus time. It keeps the diffusion operator fixed while using neural subnetworks to learn the unknown reaction terms that describe how cell populations change. Time-lapse microscopy images of lung cancer cells are preprocessed and supplied to the network, which is trained using penalties that enforce consistency with the governing structure, after which symbolic regression extracts explicit equations. If correct, this turns raw video observations into interpretable mathematical models without needing to specify every term in advance. A sympathetic reader would care because the method offers a practical route to analytic discovery for complex living systems that evolve in space and time.

Core claim

The authors demonstrate that BINNs can be extended to 2D+t reaction-diffusion systems inside a PINN framework that combines data preprocessing, BINN-based equation learning, and symbolic regression post-processing, successfully recovering governing models for lung cancer cell population dynamics from experimental time-lapse microscopy observations.

What carries the argument

Biologically-informed neural networks that preserve the known reaction-diffusion differential operator structure while learning constitutive terms through trainable neural subnetworks, enforced by soft residual penalties and followed by symbolic regression for closed-form discovery.

Load-bearing premise

The reaction-diffusion differential operator structure is an appropriate model for the biological system and the data plus soft residual penalties are sufficient for the neural subnetworks to accurately identify the constitutive terms.

What would settle it

If the equations discovered from the lung cancer cell videos produce population dynamics that fail to match held-out experimental observations or known validated models of cell growth, the claim of successful recovery would be disproved.

Figures

Figures reproduced from arXiv: 2604.18548 by Christian Olesen, Dagim S. Tadele, Jodie A. Cochrane, John T. Nardini, Sara Hamis, William Lavery.

Figure 1
Figure 1. Figure 1: PINN framework developed in this work (top) with the BINN [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Pipeline for applying the PINN framework on experimental microscopy data. [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Binned density data udata,1 and density predictions uˆ1 for replicate 1. (a) Density data across both spatial dimensions at t0 = 0. (b) Comparison of data and predictions at fixed x2 = ˜x2 = 0.143 mm for t0 and tf = 52 hours. The solid line indicates the mean across five training–validation splits, with the shaded region showing the min-max range. replicate 2 reaches high plate coverage earlier than the ot… view at source ↗
Figure 4
Figure 4. Figure 4: Growth and diffusion predictions across five training-validation (TV) splits for each biological replicate. (a) Histograms of densities used to define [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Total cell count trajectories. Observed counts [PITH_FULL_IMAGE:figures/full_fig_p006_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: BINN training curves for replicate 1. (a) Data loss history. Trained [PITH_FULL_IMAGE:figures/full_fig_p007_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: displays spatio-temporal cell density data from the microscopy images. For each replicate, τ is defined as the first time point at which at least one bin exceeds a cell density of 1000 mm2 /day. Here, τ = 24 h for all three replicates. t = τ t = τ+24 h Repl. 1 Repl. 2 Repl. 3 [PITH_FULL_IMAGE:figures/full_fig_p008_7.png] view at source ↗
read the original abstract

Physics-informed neural networks (PINNs) provide a powerful framework for learning governing equations of dynamical systems from data. Biologically-informed neural networks (BINNs) are a variant of PINNs that preserve the known differential operator structure (e.g., reaction-diffusion) while learning constitutive terms via trainable neural subnetworks, enforced through soft residual penalties. Existing BINN studies are limited to $1\mathrm{D}{+}t$ reaction-diffusion systems and focus on forward prediction, using the governing partial differential equation as a regulariser rather than an explicit identification target. Here, we extend BINNs to $2\mathrm{D}{+}t$ systems within a PINN framework that combines data preprocessing, BINN-based equation learning, and symbolic regression post-processing for closed-form equation discovery. We demonstrate the framework's real-world applicability by learning the governing equations of lung cancer cell population dynamics from time-lapse microscopy data, recovering $2\mathrm{D}{+}t$ reaction-diffusion models from experimental observations. The proposed framework is readily applicable to other spatio-temporal systems, providing a practical and interpretable tool for fast analytic equation discovery from data.

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

2 major / 3 minor

Summary. The paper extends Biologically-Informed Neural Networks (BINNs) from 1D+t to 2D+t reaction-diffusion systems within a PINN framework. It integrates data preprocessing, neural subnetworks to learn constitutive terms under soft residual penalties that preserve the known differential operator structure, and symbolic regression post-processing to obtain closed-form equations. The central demonstration is the recovery of interpretable 2D+t reaction-diffusion models for lung cancer cell population dynamics from time-lapse microscopy data.

Significance. If the empirical recovery holds with adequate validation, the work supplies a practical, interpretable pipeline for analytic equation discovery in spatio-temporal biological systems. It moves BINN methods beyond forward prediction and 1D settings toward real experimental data, with potential applicability to other reaction-diffusion problems where the operator structure is known a priori.

major comments (2)
  1. [§4.3, §5.2] §4.3 and §5.2: The recovery of the governing equations from experimental data is presented primarily through visual agreement and post-processed symbolic forms, but no quantitative metrics (e.g., residual norms, parameter recovery error on held-out data, or comparison to synthetic benchmarks) are reported for the 2D+t case. This weakens the claim that the neural subnetworks accurately identify the constitutive terms under the soft penalties.
  2. [§3.1, Eq. (5)–(7)] §3.1, Eq. (5)–(7): The BINN loss combines data fidelity with soft residual penalties whose weighting factors are free hyperparameters. The manuscript does not include an ablation or sensitivity analysis showing that the recovered symbolic equations remain stable across reasonable choices of these weights, which is load-bearing for the identification claim.
minor comments (3)
  1. [Abstract, §1] The abstract and introduction would benefit from a concise statement of the quantitative validation metrics used in the experimental section.
  2. [§2, §3] Notation for the 2D+t spatial-temporal coordinates and the neural subnetwork outputs should be introduced once and used consistently across equations and figures.
  3. [§5] Figure captions for the microscopy results should explicitly state the time interval and spatial resolution of the input data.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive review and the recommendation of minor revision. The comments identify key opportunities to strengthen the empirical validation of the equation recovery pipeline. We address each major comment below and will revise the manuscript accordingly.

read point-by-point responses

  1. Referee: [§4.3, §5.2] §4.3 and §5.2: The recovery of the governing equations from experimental data is presented primarily through visual agreement and post-processed symbolic forms, but no quantitative metrics (e.g., residual norms, parameter recovery error on held-out data, or comparison to synthetic benchmarks) are reported for the 2D+t case. This weakens the claim that the neural subnetworks accurately identify the constitutive terms under the soft penalties.

    Authors: We agree that quantitative metrics are needed to more rigorously support the identification claims. In the revised manuscript we will add (i) residual norm evaluations on held-out experimental frames, (ii) parameter recovery error statistics obtained by applying the same pipeline to synthetic 2D+t data generated from known reaction-diffusion models, and (iii) cross-validation error across multiple data splits. These additions will be placed in §4.3 and §5.2 alongside the existing visual and symbolic results. revision: yes

  2. Referee: [§3.1, Eq. (5)–(7)] §3.1, Eq. (5)–(7): The BINN loss combines data fidelity with soft residual penalties whose weighting factors are free hyperparameters. The manuscript does not include an ablation or sensitivity analysis showing that the recovered symbolic equations remain stable across reasonable choices of these weights, which is load-bearing for the identification claim.

    Authors: We acknowledge that stability with respect to the loss-weighting hyperparameters is essential for the reliability of the identification procedure. In the revised manuscript we will include an ablation study in §3.1 that systematically varies the relative weights of the data-fidelity and residual-penalty terms over a representative range (e.g., 0.1–10) and demonstrates that the downstream symbolic regression yields consistent equation structures and parameter values. The results will be summarized in a new table or figure. revision: yes

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper extends existing BINN methods to 2D+t systems by assuming the reaction-diffusion differential operator structure as known, using neural subnetworks to learn constitutive terms enforced by soft residual penalties, and applying symbolic regression post-processing to recover closed-form expressions from data. The central claim is an application to lung cancer cell microscopy data for model recovery. No derivation step reduces a claimed prediction or result to its own inputs by construction, and no load-bearing uniqueness or ansatz is imported via self-citation. The chain is self-contained with independent content from the data and standard PINN regularization.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central claim rests on the assumption that biological cell dynamics obey a reaction-diffusion PDE with a known differential operator whose reaction terms can be learned via neural subnetworks; no free parameters or invented entities are explicitly named in the abstract.

free parameters (1)
  • BINN loss weighting factors
    Typical soft-constraint weights balancing data fidelity and PDE residual in PINN-style training; not quantified in the abstract.
axioms (1)
  • domain assumption The system obeys a reaction-diffusion PDE with known differential operator structure.
    Explicitly invoked in the abstract when describing BINNs that preserve the known differential operator structure while learning constitutive terms.

pith-pipeline@v0.9.0 · 5523 in / 1327 out tokens · 43665 ms · 2026-05-10T05:14:43.735852+00:00 · methodology

discussion (0)

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