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arxiv: 2606.13475 · v1 · pith:7S3GPJJQnew · submitted 2026-06-11 · 🧬 q-bio.QM · q-bio.PE

A likelihood-based framework for simultaneously learning both noise and growth dynamics using biologically-informed neural networks

Rebecca M. Crossley , Ruth E. Baker This is my paper

Pith reviewed 2026-06-27 04:53 UTC · model grok-4.3

classification 🧬 q-bio.QM q-bio.PE
keywords biologically-informed neural networksnoise modelingpopulation growthlikelihood-based learningheteroscedastic noisemechanistic modelsneural ordinary differential equations
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The pith

An extended neural network framework learns both population growth laws and the structure of biological noise from data.

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

The paper presents a likelihood-based extension to Biologically-Informed Neural Networks that simultaneously optimizes for underlying mechanistic dynamics and a learnable noise model. Standard approaches assume constant Gaussian noise, but the new method discovers heteroscedastic structure directly from observations. In population growth examples the framework recovers the true noise pattern and produces more accurate forecasts of the growth law itself. This matters because many biological datasets contain structured variability that, if modeled, sharpens inference about the deterministic rules. The work positions the approach as a general template for mechanistic neural networks.

Core claim

Using population growth as an example, the framework accurately recovers the underlying noise structure and improves predictions of the underlying growth laws compared to existing approaches. This establishes a general likelihood-based framework for jointly learning dynamics and heteroscedastic noise within mechanistic neural network approaches.

What carries the argument

A likelihood-based extension to Biologically-Informed Neural Networks that adds a learnable noise model optimized jointly with the dynamics.

If this is right

  • The method recovers the true noise structure from sparse observations of population growth.
  • Predictions of the growth law improve relative to models that assume constant noise.
  • The same joint-learning procedure applies to other mechanistic neural-network settings beyond growth models.
  • Noise discovery becomes part of the same training loop rather than a separate post-processing step.

Where Pith is reading between the lines

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

  • If the noise model is recovered reliably, experimenters could use it to design sampling strategies that reduce uncertainty in future data collection.
  • The approach might allow direct comparison of competing mechanistic hypotheses by letting each hypothesis carry its own best noise model.
  • Extensions to spatial or multi-population systems would test whether the same joint optimization remains stable when dimensionality increases.

Load-bearing premise

The joint optimization of dynamics and noise parameters remains identifiable and does not suffer from trade-offs that allow multiple equally good explanations of the same data.

What would settle it

A dataset in which two different pairs of dynamics and noise models achieve equally low likelihood on the observations would show that unique recovery is not guaranteed.

Figures

Figures reproduced from arXiv: 2606.13475 by Rebecca M. Crossley, Ruth E. Baker.

Figure 1
Figure 1. Figure 1: Plots demonstrating the recovery of population dynamics, growth laws, and noise struc [PITH_FULL_IMAGE:figures/full_fig_p014_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Left: learned variance σ 2 (u) as a function of the population density, u, for ten replicated synthetic datasets generated from the logistic, Gompertz, and Richards’ models (Section 3) with parameters r, K, u0, and ν as specified in Section 3.1. Noise was added according to Equation (3), using α = 0 (additive, blue), α = 0.5 (intermediate, green), and α = 1 (multiplicative, red). For clarity, the logistic,… view at source ↗
Figure 3
Figure 3. Figure 3: Plot showing the calibration of predictive uncertainty for ten replicated synthetic datasets [PITH_FULL_IMAGE:figures/full_fig_p017_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Plot showing the distribution of the root mean squared error between the true underlying [PITH_FULL_IMAGE:figures/full_fig_p019_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Application of the NLL–BINN framework to coral re-growth data from two reef sites near [PITH_FULL_IMAGE:figures/full_fig_p021_5.png] view at source ↗
read the original abstract

In recent years, neural ordinary differential equation frameworks such as Biologically-Informed Neural Networks (BINNs) have shown promise for learning mechanistic laws from sparse data. However, most existing approaches implicitly assume homoscedastic Gaussian noise, and therefore do not account for potentially meaningful structure in biological variability. Here, we present an extension to the existing BINNs framework that includes a learnable noise model, allowing discovery of the noise model directly from data. Using population growth as an example, we demonstrate that the framework accurately recovers the underlying noise structure and improves predictions of the underlying growth laws compared to existing approaches. As such, this work establishes a general likelihood-based framework for jointly learning dynamics and heteroscedastic noise within mechanistic neural network approaches.

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

1 major / 1 minor

Summary. The manuscript extends Biologically-Informed Neural Networks (BINNs) with a likelihood-based framework that jointly optimizes mechanistic dynamics and a learnable heteroscedastic noise model. Using simulated population-growth data as the running example, it claims that the approach recovers the true noise structure and yields improved predictions of the underlying growth laws relative to homoscedastic baselines.

Significance. If the joint optimization is identifiable, the framework would address a genuine limitation of existing BINN-style methods by allowing discovery of structured biological variability. The use of simulated data to test recovery is a positive step, but the absence of any identifiability analysis or ablation that isolates dynamics from noise reduces the strength of the central claim.

major comments (1)
  1. [Methods / Results] The central claim that the framework 'accurately recovers the underlying noise structure' (abstract) rests on the assumption that maximizing the joint likelihood separates dynamics from noise without compensatory trade-offs. No identifiability analysis, Hessian diagnostic, or ablation that fixes one component while varying the other is provided; different (dynamics, noise) pairs can produce nearly identical marginal likelihoods under sparse sampling or flexible state-dependent variance, which directly threatens the reported improvement over homoscedastic baselines.
minor comments (1)
  1. The abstract and introduction would benefit from explicit statements of the noise-model families considered (e.g., state-dependent variance forms) and the quantitative recovery metrics (e.g., parameter error, predictive log-likelihood) used to claim superiority.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their constructive comments. We address the major concern regarding the identifiability of the joint optimization below.

read point-by-point responses

  1. Referee: [Methods / Results] The central claim that the framework 'accurately recovers the underlying noise structure' (abstract) rests on the assumption that maximizing the joint likelihood separates dynamics from noise without compensatory trade-offs. No identifiability analysis, Hessian diagnostic, or ablation that fixes one component while varying the other is provided; different (dynamics, noise) pairs can produce nearly identical marginal likelihoods under sparse sampling or flexible state-dependent variance, which directly threatens the reported improvement over homoscedastic baselines.

    Authors: We appreciate the referee's point on the need for identifiability analysis. In our work, we used simulated data with known ground truth to demonstrate that the framework recovers both the correct growth dynamics and the heteroscedastic noise structure. Multiple random initializations consistently converged to the true parameters, providing empirical evidence of separation. Nevertheless, we agree that a more rigorous analysis would strengthen the manuscript. In the revision, we will add a discussion of identifiability considerations and include an ablation study where the noise model is fixed to homoscedastic while learning dynamics, and vice versa, to isolate the contributions. This will better support the claimed improvements. revision: yes

Circularity Check

0 steps flagged

No circularity: framework and recovery claims are externally validated on simulated data

full rationale

The provided abstract and context describe an extension of BINNs that jointly optimizes dynamics and a learnable heteroscedastic noise model via likelihood. No equations or steps are quoted that reduce a claimed prediction to a fitted parameter by construction, nor does any load-bearing premise rely on a self-citation chain. Recovery of noise structure and growth laws is demonstrated on simulated population-growth data against homoscedastic baselines, constituting an independent benchmark rather than a renaming or self-definition. The identifiability assumption is a modeling risk but does not trigger any of the enumerated circularity patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review supplies no concrete free parameters, axioms, or invented entities; the central claim rests on the unstated assumption that joint likelihood optimization separates dynamics from noise.

pith-pipeline@v0.9.1-grok · 5658 in / 891 out tokens · 21926 ms · 2026-06-27T04:53:35.598105+00:00 · methodology

discussion (0)

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