Leveraging Biokinetic Knowledge Priors for Data-Scarce Bioprocess Modeling

# Leveraging Biokinetic Knowledge Priors for Data-Scarce Bioprocess Modeling
Source: [https://arxiv.org/html/2607.20539](https://arxiv.org/html/2607.20539)
###### Abstract

While deep learning has accelerated drug discovery, its impact on biomanufacturing has been considerably more limited\. The reason is data scarcity\. Bioreactor experiments are high\-cost, take days to weeks, and are rarely shared in public form, leaving each research work with only a handful of experiments\. The domain itself, however, is rich in prior knowledge\. Biokinetic ordinary differential equation \(ODE\) models have described microbial growth for decades, yet how to inject this knowledge into a neural network has not been studied systematically\.

We present the first systematic study of how to inject this ODE knowledge into a neural network, comparing a*data\-level prior*that pre\-trains a generic decoder on simulated ODE curves against an*architecture\-level prior*that embeds the ODE inside the decoder\. Both consistently outperform no\-prior baselines across1111datasets and77microbial species\. Our central finding is that the two are*substitutable*\. A generic decoder pre\-trained on simulation matches a fully bio\-structured decoder trained on real data\. Simulation pre\-training therefore offers a simple, data\-efficient recipe for deep learning under bioprocess data scarcity\.

Bioprocess, Biokinetic ODE, Simulation Pre\-training, AI for Science

## 1Introduction

![Refer to caption](https://arxiv.org/html/2607.20539v1/x1.png)Figure 1:Task overview and the two biokinetic prior\-injection channels compared in this paper\.\(a\) Task:given environmental conditions𝐞\\mathbf\{e\}\(e\.g\., temperature, pH, substrate, medium\) and an optional set of early observations𝒞\\mathcal\{C\}, predict the bioreactor state trajectoryy^​\(t\)\\hat\{y\}\(t\)\(cell density or an equivalent measurement\)\.\(b\) Two injection channels:simulation pre\-traininggenerates synthetic curves from parameterized biokinetic ODEs and uses them to pre\-train a generic decoder, while anarchitecture\-level priorembeds the ODE directly in the decoder’s forward pass\. Section[3](https://arxiv.org/html/2607.20539#S3)formalizes both channels on a shared encoder–decoder backbone\.#### Bottleneck in bioprocess AI

Deep learning has driven major advances across the early stages of drug discovery\(Vamathevanet al\.,[2019](https://arxiv.org/html/2607.20539#bib.bib9); Jiménez\-Lunaet al\.,[2020](https://arxiv.org/html/2607.20539#bib.bib58)\)\. Recent systems for biomolecular structure prediction\(Jumperet al\.,[2021](https://arxiv.org/html/2607.20539#bib.bib28); Abramsonet al\.,[2024](https://arxiv.org/html/2607.20539#bib.bib54); Linet al\.,[2023](https://arxiv.org/html/2607.20539#bib.bib55)\), de novo protein design\(Watsonet al\.,[2023](https://arxiv.org/html/2607.20539#bib.bib56)\), antibiotic discovery\(Stokeset al\.,[2020](https://arxiv.org/html/2607.20539#bib.bib53)\), and cellular simulation\(Bunneet al\.,[2024](https://arxiv.org/html/2607.20539#bib.bib44)\)now contribute meaningfully to molecular design, target identification, and lead optimization\.

The downstream stage of biomanufacturing, in which candidate molecules must be produced in bioreactors at scale, has by contrast attracted far less ML attention\(Mowbrayet al\.,[2021](https://arxiv.org/html/2607.20539#bib.bib45); Helleckeset al\.,[2023](https://arxiv.org/html/2607.20539#bib.bib46)\)\. A bioreactor experiment typically takes days to weeks and consumes substantial reagents, and every new product or strain demands re\-tuning of cultivation conditions through trial and error\(Faridet al\.,[2020](https://arxiv.org/html/2607.20539#bib.bib47)\)\. Predicting bioreactor states such as cell density, substrate concentration, and product titer before an experiment is run would directly reduce cycle time and sample consumption, and would establish a substantive role for machine learning at this stage\.

#### Research gap

Three structural factors keep bioprocess modeling under\-investigated by the ML community\.Data scarcity by experimental nature:each cultivation is costly in both wall\-clock time and reagents, so the data any single group can accumulate is intrinsically limited\(Barón Díaz and others,[2025](https://arxiv.org/html/2607.20539#bib.bib48)\)\.Industrial confidentiality:process data are treated as trade secrets, and very few datasets ever reach a public, curated form\(Smiatek and others,[2024](https://arxiv.org/html/2607.20539#bib.bib49)\)\.Method\-level fragmentation:as a consequence, datasets across groups serve different objectives, and direct method\-to\-method comparison is rarely performed; results from one group seldom transfer to data from other groups\(Khanalet al\.,[2024](https://arxiv.org/html/2607.20539#bib.bib50)\)\. Together, these factors explain why machine learning for bioprocess prediction has progressed unevenly to date\. Promising single\-dataset case studies exist, but unified empirical baselines are rare\.

#### Biokinetic ODE model

Microbial growth dynamics, in contrast, have been described by biokinetic ODE models for several decades\(Monod,[1949](https://arxiv.org/html/2607.20539#bib.bib1); Zwieteringet al\.,[1990](https://arxiv.org/html/2607.20539#bib.bib4); Baranyi and Roberts,[1994](https://arxiv.org/html/2607.20539#bib.bib2); Rossoet al\.,[1995](https://arxiv.org/html/2607.20539#bib.bib6)\)\. These models describe cell\-level dynamics in closed form\. Once organism\-specific parameters are fitted from the literature, simulation curves can be synthesized for any organism\. This makes biokinetic ODEs a rare resource in a data\-scarce setting: prior knowledge that is mathematically precise, organism\-portable, and freely available\. What remains open is a single design question:*how should this knowledge be injected into a neural network?*

#### Two prior\-injection channels

We consider two orthogonal channels for injecting biokinetic ODE knowledge into a neural network, following the broader taxonomy of bias injection in physics\-informed machine learning\(Karniadakiset al\.,[2021](https://arxiv.org/html/2607.20539#bib.bib51)\)\. The first is*simulation pre\-training*\(a data\-level prior\), in which we generate large synthetic datasets from biokinetic ODEs, pre\-train a decoder, and fine\-tune on real data; pre\-training on synthetic data has been a successful strategy in other data\-scarce regimes\(Hollmannet al\.,[2023](https://arxiv.org/html/2607.20539#bib.bib52)\)\. The second is an*architecture\-level prior*, in which the ODE itself is embedded in the model’s forward pass as a template, hybrid, or Neural ODE backbone\.

Prior work typically commits to one channel or the other\. Whether the two routes are complementary, substitutable, or one is strictly stronger has not been studied, even though they target the same underlying knowledge\. This paper is, to our knowledge, the first empirical study to compare the two channels under a single task, dataset suite, and shared backbone\. We also answer the practical question of*how to construct an effective simulation dataset*\(e\.g\., which ODE family, which parameter distribution, and how much data\)\. Figure[1](https://arxiv.org/html/2607.20539#S1.F1)summarizes the task and the family of models we compare\.

Our contributions are summarized as follows\.

1. 1\.To our knowledge, this is the first empirical study that compares two channels for injecting biokinetic domain knowledge into neural networks, namely*simulation pre\-training*and*architecture\-level priors*, under a single task, shared backbone, and unified evaluation, across1111datasets and77microbial species\.
2. 2\.Across these datasets, biokinetic priors consistently improve performance: both channels improve over no\-prior baselines, and the improvement scales monotonically with prior intensity\.
3. 3\.Simulation pre\-training is the more effective channel: a generic neural decoder paired with it matches a fully bio\-structured decoder, demonstrating that the two channels are substitutable and that simulation is the more data\-efficient route\.
4. 4\.Finally, we offer a practical recipe for constructing simulation datasets, with three findings: random simulation fails, composite\-biokinetic simulation with broad parameter sampling performs best, and pre\-training is at least as effective as joint training\.

## 2Related Work

#### Data\-driven ML/DL for bioprocess prediction

Classical ML, such as PLSR, SVR, Gaussian processes, and XGBoost\(Peng and others,[2025](https://arxiv.org/html/2607.20539#bib.bib12); Khuat and others,[2025](https://arxiv.org/html/2607.20539#bib.bib13)\), and feed\-forward MLPs \(e\.g\., for CHO mAb titer\(Richter and others,[2025](https://arxiv.org/html/2607.20539#bib.bib11)\)\) have been applied steadily to bioprocess prediction\. Sequence models such as the LSTM ofBonanni and others \([2023](https://arxiv.org/html/2607.20539#bib.bib29)\)forE\. coliOD600 forecasting \(the basis of our GRU baseline\) and Deep Set or autoencoder representations\(Borisyak and others,[2023](https://arxiv.org/html/2607.20539#bib.bib14); Baig and others,[2023](https://arxiv.org/html/2607.20539#bib.bib30)\)extend the same data\-driven paradigm\. These works learn the input–output mapping as a black box; biokinetic domain knowledge is rarely integrated explicitly\.

#### Biokinetic models

Mechanistic ODEs for microbial growth date back several decades\. Monod’s saturation kinetics\(Monod,[1949](https://arxiv.org/html/2607.20539#bib.bib1)\), Gompertz/Zwietering\(Zwieteringet al\.,[1990](https://arxiv.org/html/2607.20539#bib.bib4)\), Baranyi & Roberts lag dynamics\(Baranyi and Roberts,[1994](https://arxiv.org/html/2607.20539#bib.bib2)\), and Luedeking–Piret product formation\(Luedeking and Piret,[1959](https://arxiv.org/html/2607.20539#bib.bib3)\)together describe biomass, substrate, and product dynamics through coupled mass\-balance ODEs, and scale up to large process simulators such as IndPenSim\(Goldricket al\.,[2015](https://arxiv.org/html/2607.20539#bib.bib8); Goldrick and others,[2019](https://arxiv.org/html/2607.20539#bib.bib43)\)\. Their dominant application is in process control validation\(Li and others,[2024](https://arxiv.org/html/2607.20539#bib.bib40); Petsagkourakis and others,[2020](https://arxiv.org/html/2607.20539#bib.bib39)\)or soft\-sensor benchmarking\(Peng and others,[2025](https://arxiv.org/html/2607.20539#bib.bib12)\); the use of biokinetic ODEs as a learning prior for neural networks has received little systematic attention\.

#### Biokinetic\-informed deep learning

Existing approaches divide along three lines\.PINN \(loss\-level\)adds the biokinetic ODE residual as an auxiliary loss\(Adebar and others,[2025](https://arxiv.org/html/2607.20539#bib.bib18); Zhu and others,[2026](https://arxiv.org/html/2607.20539#bib.bib41); Kusters and others,[2025](https://arxiv.org/html/2607.20539#bib.bib20)\)\.Hybrid \(architecture\-level\)replaces reaction kinetics with a neural network on top of mass\-balance ODEs\(Pinto and others,[2022](https://arxiv.org/html/2607.20539#bib.bib15); Ramos and others,[2024](https://arxiv.org/html/2607.20539#bib.bib16)\), or estimates time\-varying parameters with a neural net\(Shah and others,[2022](https://arxiv.org/html/2607.20539#bib.bib17); Riezzo and others,[2025](https://arxiv.org/html/2607.20539#bib.bib42)\)\.Neural ODE/UDE \(solver\-level\)places an ODE solver inside the forward pass\(Bangi and Kwon,[2022](https://arxiv.org/html/2607.20539#bib.bib22); Chiu and others,[2024](https://arxiv.org/html/2607.20539#bib.bib21)\)\. All of these works apply a single integration route to a single process; the gap we address is a systematic comparison of the two core channels \(data\-level simulation pre\-training and architecture\-level integration\) on a single task, dataset suite, and shared backbone\.

## 3Method

### 3\.1Bioreactor state prediction

Public bioprocess datasets are dominated by microbial growth experiments, and fed\-batch records are very rarely shared in a curated form, so we restrict our task formulation to batch microbial cultivation throughout the paper\. We formalize bioreactor state prediction in this regime as a conditional trajectory regression task\. Each cultivation runiiis described by a static*environmental condition*vector𝐞i∈ℝde\\mathbf\{e\}\_\{i\}\\in\\mathbb\{R\}^\{d\_\{e\}\}, encoding cultivation parameters such as temperature, pH, initial substrate concentration, and medium composition\. Optionally, a set of*context observations*𝒞i=\{\(tk,yi,k\)\}k=1Ki\\mathcal\{C\}\_\{i\}=\\\{\(t\_\{k\},y\_\{i,k\}\)\\\}\_\{k=1\}^\{K\_\{i\}\}provides up toKiK\_\{i\}early measurements collected during the same cultivation;𝒞i\\mathcal\{C\}\_\{i\}may be empty\. The model predicts the bioreactor statey^i​\(t\)∈ℝ\\hat\{y\}\_\{i\}\(t\)\\in\\mathbb\{R\}at any query timett, whereyydenotes cell density or an equivalent measurement such as OD600\. Supervision is provided by ground\-truth observationsyi​\(tj\)y\_\{i\}\(t\_\{j\}\)on the time grid𝒯i=\{t1,…,tTi\}\\mathcal\{T\}\_\{i\}=\\\{t\_\{1\},\\ldots,t\_\{T\_\{i\}\}\\\}at which the cultivation was sampled\.

Every model in this paper factorizes the predictor into the same two\-stage mapping,

𝐳i=Encoder​\(𝐞i,𝒞i\),y^i​\(t\)=Decoder​\(𝐳i,t\),\\mathbf\{z\}\_\{i\}\\;=\\;\\mathrm\{Encoder\}\\\!\\left\(\\mathbf\{e\}\_\{i\},\\,\\mathcal\{C\}\_\{i\}\\right\),\\qquad\\hat\{y\}\_\{i\}\(t\)\\;=\\;\\mathrm\{Decoder\}\\\!\\left\(\\mathbf\{z\}\_\{i\},\\,t\\right\),\(1\)where𝐳i∈ℝdz\\mathbf\{z\}\_\{i\}\\in\\mathbb\{R\}^\{d\_\{z\}\}is a per\-cultivation latent representation produced by a shared encoder \(EnvEncoder\+\+ContextEncoder; see Appendix[C](https://arxiv.org/html/2607.20539#A3)\)\. Only the decoder differs from one baseline to another, which isolates the architectural prior as the only source of variation across models\.

Models are trained against the trajectory regression loss

ℒi=1Ti​∑j=1Tiℓ​\(y^i​\(tj\),yi​\(tj\)\),\\mathcal\{L\}\_\{i\}\\;=\\;\\frac\{1\}\{T\_\{i\}\}\\sum\_\{j=1\}^\{T\_\{i\}\}\\ell\\\!\\left\(\\hat\{y\}\_\{i\}\(t\_\{j\}\),\\;y\_\{i\}\(t\_\{j\}\)\\right\),\(2\)whereℓ​\(⋅,⋅\)\\ell\(\\cdot,\\cdot\)is a per\-point regression loss\. The simplest concrete instance is the squared errorℓ​\(y^,y\)=\(y^−y\)2\\ell\(\\hat\{y\},y\)=\(\\hat\{y\}\-y\)^\{2\}, which yields the mean\-squared\-error \(MSE\) loss\. All baselines compared in this paper share this trajectory regression form; the specific choice ofℓ\\elland the evaluation metrics are defined in Section[4](https://arxiv.org/html/2607.20539#S4)\.

### 3\.2Biokinetic prior–injected models

#### Component\-based hybrid family

Every prior\-injected baseline in this paper fits a unified template: a biokinetic prior plus a small neural correction,

y^i​\(t\)=fprior​\(t;𝜽i\)\+ε⋅gθ​\(state,t,𝐳i\),\\hat\{y\}\_\{i\}\(t\)\\;=\\;f\_\{\\mathrm\{prior\}\}\\\!\\big\(t;\\,\\boldsymbol\{\\theta\}\_\{i\}\\big\)\\;\+\\;\\varepsilon\\cdot g\_\{\\theta\}\\\!\\big\(\\mathrm\{state\},\\,t,\\,\\mathbf\{z\}\_\{i\}\\big\),\(3\)where𝜽i=ParamHead​\(𝐳i\)\\boldsymbol\{\\theta\}\_\{i\}=\\mathrm\{ParamHead\}\(\\mathbf\{z\}\_\{i\}\)andε∈\{0\.01,0\.1\}\\varepsilon\\in\\\{0\.01,0\.1\\\}\. Herefpriorf\_\{\\mathrm\{prior\}\}is a biokinetic template \(a closed form or an ODE solution\),gθg\_\{\\theta\}is a small neural residual, and𝜽i\\boldsymbol\{\\theta\}\_\{i\}collects the organism\-specific dynamic parameters \(e\.g\.,μmax\\mu\_\{\\max\},KsK\_\{s\},KK,kdk\_\{d\}\) emitted by ParamHead from the latent𝐳i\\mathbf\{z\}\_\{i\}\. The scalarε\\varepsiloncontrols the influence of the neural correction: a smallε\\varepsilonfavors the prior, while a largeε\\varepsilonfavors the neural network\. Within this template we instantiate four representative architectures whose prior intensity increases from left to right\.

#### MLP

We start at the no\-prior end of the spectrum\. This baseline is thefprior≡0f\_\{\\mathrm\{prior\}\}\\equiv 0limit of[Equation3](https://arxiv.org/html/2607.20539#S3.E3), in which the decoder is a44\-layer fully\-connected ReLU network with no ODE structure:

y^i​\(t\)=MLP​\(𝐳i,t\)\.\\hat\{y\}\_\{i\}\(t\)\\;=\\;\\mathrm\{MLP\}\\\!\\big\(\\mathbf\{z\}\_\{i\},\\,t\\big\)\.\(4\)The MLP carries no biokinetic structure and acts as the no\-prior reference point in our comparison\.

#### PINN

At a higher level of prior intensity, we retain the same backbone and incorporate the biokinetic ODE into the loss instead of the architecture\. Building on the physics\-informed neural network framework\(Raissiet al\.,[2019](https://arxiv.org/html/2607.20539#bib.bib57)\)and following its bioprocess instantiation inAdebar and others \([2025](https://arxiv.org/html/2607.20539#bib.bib18)\), the PINN baseline uses the same MLP backbone as above and adds an autograd\-based logistic\-residual term to the loss:

ℒiPINN=\\displaystyle\\mathcal\{L\}^\{\\mathrm\{PINN\}\}\_\{i\}\\;=\\;\(1−λ\)​ℒiMSE\+λTi​∑j=1Ti\\displaystyle\(1\-\\lambda\)\\,\\mathcal\{L\}\_\{i\}^\{\\mathrm\{MSE\}\}\\;\+\\;\\frac\{\\lambda\}\{T\_\{i\}\}\\sum\_\{j=1\}^\{T\_\{i\}\}\|d​y^id​t​\(tj\)−μ​y^i​\(tj\)​\(1−y^i​\(tj\)K\)\|2\.\\displaystyle\\Big\|\\tfrac\{d\\hat\{y\}\_\{i\}\}\{dt\}\(t\_\{j\}\)\-\\mu\\,\\hat\{y\}\_\{i\}\(t\_\{j\}\)\\\!\\left\(1\-\\tfrac\{\\hat\{y\}\_\{i\}\(t\_\{j\}\)\}\{K\}\\right\)\\Big\|^\{2\}\.\(5\)The derivatived​y^i/d​td\\hat\{y\}\_\{i\}/dtis evaluated by autograd, the trade\-off coefficient is fixed atλ=0\.1\\lambda=0\.1, and the dataset\-level\(μ,K\)\(\\mu,K\)are taken from the same Gompertz fits used by ODE\-Fit so that the loss\-channel and the template\-channel see the same kinetic prior\. The architecture is unchanged, but the model is now regularized toward biokinetic dynamics during training\.

#### Hybrid\-NeuralODE

Pushing the prior further into the architecture, we embed the ODE itself inside the forward pass and let a small neural network correct it\(Chenet al\.,[2018](https://arxiv.org/html/2607.20539#bib.bib25)\)\. FollowingBangi and Kwon \([2022](https://arxiv.org/html/2607.20539#bib.bib22)\), this baseline integrates a Baranyi two\-state system,

d​Nid​t\\displaystyle\\frac\{dN\_\{i\}\}\{dt\}=μ⋅qi1\+qi⋅\(1−NiK\)⋅Ni\+ε​gθ​\(Ni,t,𝐳i\),\\displaystyle=\\mu\\cdot\\frac\{q\_\{i\}\}\{1\+q\_\{i\}\}\\cdot\\Big\(1\-\\tfrac\{N\_\{i\}\}\{K\}\\Big\)\\cdot N\_\{i\}\+\\varepsilon\\,g\_\{\\theta\}\(N\_\{i\},t,\\mathbf\{z\}\_\{i\}\),d​qid​t\\displaystyle\\frac\{dq\_\{i\}\}\{dt\}=μ⋅qi,\\displaystyle=\\mu\\cdot q\_\{i\},\(6\)whereNiN\_\{i\}is cell density,qiq\_\{i\}is the Baranyi lag adaptation state, andμ\\muis treated as a constant \(substrate dynamics are not modeled here\)\. The model output is now an ODE solution, and the neural correctiongθg\_\{\\theta\}modulates that solution only marginally\.

#### BioStruct\-ODE family

At the strong\-prior end of the spectrum, we introduce the BioStruct\-ODE family, which adopts the Monod–Baranyi system as the model’s backbone\. The system tracks four states explicitly, namely cell densityNiN\_\{i\}, substrateSiS\_\{i\}, productPiP\_\{i\}, and lag adaptationqiq\_\{i\}, and includes an explicit death term:

d​Nid​t\\displaystyle\\frac\{dN\_\{i\}\}\{dt\}=μ​\(Si\)⋅qi1\+qi⋅\(1−NiK\)​Ni−kd​Ni\+ε​gθ,\\displaystyle=\\mu\(S\_\{i\}\)\\\!\\cdot\\\!\\frac\{q\_\{i\}\}\{1\+q\_\{i\}\}\\\!\\cdot\\\!\\Big\(1\-\\tfrac\{N\_\{i\}\}\{K\}\\Big\)N\_\{i\}\-k\_\{d\}N\_\{i\}\+\\varepsilon\\,g\_\{\\theta\},d​Sid​t\\displaystyle\\frac\{dS\_\{i\}\}\{dt\}=−μ​\(Si\)Yx​s​Ni\+ε​gθ,\\displaystyle=\-\\,\\frac\{\\mu\(S\_\{i\}\)\}\{Y\_\{xs\}\}\\,N\_\{i\}\+\\varepsilon\\,g\_\{\\theta\},d​Pid​t\\displaystyle\\frac\{dP\_\{i\}\}\{dt\}=α​μ​\(Si\)​Ni\+β​Ni\+ε​gθ,\\displaystyle=\\alpha\\,\\mu\(S\_\{i\}\)\\,N\_\{i\}\+\\beta\\,N\_\{i\}\+\\varepsilon\\,g\_\{\\theta\},d​qid​t\\displaystyle\\frac\{dq\_\{i\}\}\{dt\}=μ​\(Si\)⋅qi\.\\displaystyle=\\mu\(S\_\{i\}\)\\cdot q\_\{i\}\.\(7\)The growth rateμ​\(Si\)\\mu\(S\_\{i\}\)follows Monod kinetics, and the bias of ParamHead is*warm\-started*from per\-organism literature Monod fits to stabilize training \(Appendix[C](https://arxiv.org/html/2607.20539#A3)\)\. The family contains two variants\.BioStruct\-ODEuses the system above as is\.BioStruct\-ODE\-Cardinaladditionally models the dependence ofμ\\muon temperature and pH through Rosso’s cardinal envelope\(Rossoet al\.,[1995](https://arxiv.org/html/2607.20539#bib.bib6)\),

μ​\(Si,Ti,pHi\)=γ​\(Ti\)⋅γ​\(pHi\)⋅μmax⋅SiKs\+Si,\\mu\(S\_\{i\},T\_\{i\},\\mathrm\{pH\}\_\{i\}\)\\;=\\;\\gamma\(T\_\{i\}\)\\cdot\\gamma\(\\mathrm\{pH\}\_\{i\}\)\\cdot\\mu\_\{\\max\}\\cdot\\frac\{S\_\{i\}\}\{K\_\{s\}\+S\_\{i\}\},\(8\)whereγ​\(⋅\)\\gamma\(\\cdot\)is the cardinal envelope defined by the cardinal points\(Tmin,Topt,Tmax\)\(T\_\{\\min\},T\_\{\\mathrm\{opt\}\},T\_\{\\max\}\)and\(pHmin,pHopt,pHmax\)\(\\mathrm\{pH\}\_\{\\min\},\\mathrm\{pH\}\_\{\\mathrm\{opt\}\},\\mathrm\{pH\}\_\{\\max\}\)\. The Cardinal variant is applied only to datasets with an explicit environmental axis; datasets without such an axis fall back to BioStruct\-ODE automatically\.

### 3\.3Training with simulated datasets

#### Simulation data synthesis

We sample organism\-specific parameters from literature ranges and integrate the corresponding ODE family to obtain simulation curves\. Three design factors, ablated in RQ2, control the resulting synthetic dataset\.Factor A \(ODE family\)chooses among Monod, Logistic, Gompertz, Baranyi, and a Rosso\-composite form\.Factor B \(parameter sampling\)contrasts a literature\-narrow regime \(±\\pm10%\) with a broad uniform regime \(±\\pm50%\)\.Factor C \(biokinetic specificity\)arranges three regimes in increasing specificity: random Gaussian processes \(no biokinetic structure\), single\-ODE narrow, and composite\-ODE broad\. Full ODE equations and per\-organism parameter ranges are deferred to Appendix[B](https://arxiv.org/html/2607.20539#A2)\.

#### Pre\-training

We train on simulation to convergence and then fine\-tune on real data with a smaller learning rate\. The simulation budget is scaled relative to the real dataset, so that when real data is scarce, more simulation is used to cover a broader biokinetic regime; we sweep the ratio\|sim\|/\|real\|∈\{1,3,5,10,30\}\|\\mathrm\{sim\}\|/\|\\mathrm\{real\}\|\\in\\\{1,3,5,10,30\\\}\.

#### Joint training

Simulation and real samples are mixed within each batch and weighted by a curriculum coefficientα​\(t\)\\alpha\(t\)that decays from11\(simulation\-dominated\) to0\(real\-only\) as training progresses,

ℒjoint=α​\(t\)​wsim​ℒsim\+\(1−α​\(t\)\)​ℒreal,\\mathcal\{L\}\_\{\\mathrm\{joint\}\}\\;=\\;\\alpha\(t\)\\,w\_\{\\mathrm\{sim\}\}\\,\\mathcal\{L\}\_\{\\mathrm\{sim\}\}\+\(1\-\\alpha\(t\)\)\\,\\mathcal\{L\}\_\{\\mathrm\{real\}\},\(9\)withwsim∈\{0\.5,1,2,5\}w\_\{\\mathrm\{sim\}\}\\in\\\{0\.5,1,2,5\\\}\. Optimiser, schedule, loss\-term, and batch\-composition details appear in Appendix[D](https://arxiv.org/html/2607.20539#A4)\.

## 4Experiments

In this section we address four research questions\.

- •RQ1\.How do different ways of injecting an architectural biokinetic prior compare in performance?
- •RQ2\.How should simulation data be synthesized, and how should it be injected into the model \(pre\-training vs\. joint training\)?
- •RQ3\.How does test\-time context \(initial conditions only vs\. partial observations\) affect performance?
- •RQ4\.What kinds of growth\-curve errors does simulation pre\-training systematically correct?

#### Datasets

We curate1111datasets covering77bacterial species \(E\. coli,Listeria,Shigella,Staphylococcus aureus,Yersinia,Salmonella, and Pseudomonads\), comprising two batch datasets of our own\(University College London Research Data Repository,[2024](https://arxiv.org/html/2607.20539#bib.bib33); Katipoglu\-Yazan and others,[2023](https://arxiv.org/html/2607.20539#bib.bib32)\)and nine external datasets\(Faure and others,[2023](https://arxiv.org/html/2607.20539#bib.bib34); Buchanan and Phillips,[1989](https://arxiv.org/html/2607.20539#bib.bib35); Zaika and others,[1994](https://arxiv.org/html/2607.20539#bib.bib36); Eifert and others,[1997](https://arxiv.org/html/2607.20539#bib.bib37); ComBase Consortium,[2024](https://arxiv.org/html/2607.20539#bib.bib38)\)\. Per\-dataset organism, curve count, time horizon, and split are reported in Appendix[A](https://arxiv.org/html/2607.20539#A1)\.

#### Pre\-processing

OD\-native datasets \(BL21, Katipoglu23, Faure23\) are normalized in OD600 units, while ComBase data are kept in their native logCFU units\. We use a curve\-level random train/val/test split, and apply awith\_growthfilter that removes any curve whose per\-curve Monod–Baranyi NLS fit attainsR2<0\.4R^\{2\}<0\.4\. Details are in Appendix[A](https://arxiv.org/html/2607.20539#A1)\.

#### Models and implementation

All baselines share the same backbone \(EnvEncoder\+\+ContextEncoder\), and only the decoder changes\. Each model is trained with55seeds\. The default per\-point loss is shared across baselines, and PINN additionally uses the auxiliary physics\-residual term in[Equation5](https://arxiv.org/html/2607.20539#S3.E5); full loss forms are listed in Appendix[D\.1](https://arxiv.org/html/2607.20539#A4.SS1)\. We optimize with AdamW \(lr3×10−43\\times 10^\{\-4\}, weight decay10−510^\{\-5\},1010\-epoch warmup\), and use early stopping on validationR2R^\{2\}with patience55\. Further detail is in Appendices[C](https://arxiv.org/html/2607.20539#A3)and[D](https://arxiv.org/html/2607.20539#A4)\.

#### Baselines

We evaluate nine baselines\.Meanpredicts the per\-condition mean trajectory of the training set\.ODE\-Fitperforms per\-curve nonlinear least\-squares fits with Monod or Gompertz forms\(Zwieteringet al\.,[1990](https://arxiv.org/html/2607.20539#bib.bib4)\)\.MLPis a pure feed\-forward decoder\.GRUis an autoregressive recurrent decoder followingBonanni and others \([2023](https://arxiv.org/html/2607.20539#bib.bib29)\)\.PINNpairs the MLP backbone with the auxiliary physics loss ofAdebar and others \([2025](https://arxiv.org/html/2607.20539#bib.bib18)\)\.ODE\-Guidecombines a closed\-form Gompertz template with a neural residual\(Pinto and others,[2022](https://arxiv.org/html/2607.20539#bib.bib15)\)\.Hybrid\-NeuralODEcombines a mass\-balance ODE with a neural correction\(Bangi and Kwon,[2022](https://arxiv.org/html/2607.20539#bib.bib22)\)\. The two main variants of this paper areBioStruct\-ODEandBioStruct\-ODE\+Cardinal\(Section[3\.2](https://arxiv.org/html/2607.20539#S3.SS2)\); the Cardinal variant requires an explicit temperature axis in the dataset, so we evaluate it only on the77datasets that carry one, while every other baseline is evaluated on all1111\. Architectural details are in Appendix[C](https://arxiv.org/html/2607.20539#A3)\.

Table 1:Per\-model performance on the1111\-dataset suite at zero context \(RQ1\)\. Each model is trained for55seeds without simulation pre\-training; values are mean±\\pmstandard deviation, and the Rank column reports each model’s rank averaged across datasets\. The best value in every column is shown in bold\.
#### Evaluation

We report five metrics:

- •R2R^\{2\}\(primary, dimensionless\), the proportion of trajectory variance explained by the prediction;
- •AfA\_\{f\}\(Accuracy Factor,Af=10\|log10⁡\(y^/y\)\|¯A\_\{f\}=10^\{\\overline\{\|\\log\_\{10\}\(\\hat\{y\}/y\)\|\}\};Ross,[1996](https://arxiv.org/html/2607.20539#bib.bib31)\), the average fold\-change between predicted and observed values;
- •RMSE in log10 cell density, ensuring fair comparison across native units;
- •EndPntErr in OD600 at the final timepoint, with eval\-time per\-organism offsets for ComBase;
- •Rank, the average rank of each model across\(dataset,seed\)\(\\text\{dataset\},\\text\{seed\}\)pairs\.

R2R^\{2\},AfA\_\{f\}, and Rank are scale\-invariant; RMSE is computed in log space; EndPntErr is computed in OD600, where endpoints cluster at the plateau OD∼\\sim0\.3–1\.0\.

### 4\.1RQ1: Comparison of prediction models with biokinetic prior

A unified comparison of nine baselinesPrior work studies the five architectural routes \(loss\-level, template, hybrid, Neural ODE, full bio\-structured\) in isolation, on different datasets, with different encoders and evaluation protocols, which makes their relative strength impossible to infer from the published numbers\. We therefore evaluate nine baselines on the full1111\-dataset suite for55seeds with no simulation, sharing the unified backbone of Section[3](https://arxiv.org/html/2607.20539#S3)so that the only varying factor across rows is the decoder; results are reported in Table[1](https://arxiv.org/html/2607.20539#S4.T1)\.

Prior intensity orders performanceThe BioStruct\-ODE family attains the best meanR2R^\{2\}\(0\.5210\.521for the base variant and0\.5540\.554for the Cardinal variant on its seven temperature\-axis datasets\) and the best average rank \(3\.003\.00and2\.762\.76out of nine\), outperforming every other baseline\. The loss\-level prior \(PINN,R2=0\.503R^\{2\}=0\.503\) and the sequence\-bias channel \(GRU,R2=0\.494R^\{2\}=0\.494\) cluster just below the BioStruct\-ODE family\. Mean \(0\.1570\.157\), out\-of\-sample ODE\-Fit \(0\.2040\.204\), and ODE\-Guide \(0\.0900\.090\) rank at the bottom: a biokinetic template without data\-driven adaptation \(ODE\-Fit\) only barely exceeds the trivial Mean baseline, while a closed\-form prior with only a weak neural correction \(ODE\-Guide\) performs worse than the Mean baseline\.

Variance tightens, and component ablation supports itThe no\-prior MLP exhibits a per\-curve standard deviation of1\.201\.20that is dominated by a small number of difficult datasets \(notably Zaika19941994, where the unconstrained network produces strongly negativeR2R^\{2\}\), while the BioStruct\-ODE family suppresses this to about0\.210\.21\. The biokinetic component ablation in Appendix[E\.1](https://arxiv.org/html/2607.20539#A5.SS1)\(Table[8](https://arxiv.org/html/2607.20539#A5.T8)\) supports this attribution: removing the death term, the warm\-start, or switching the template from Monod–Baranyi to Gompertz each incurs a small but consistent loss in bothR2R^\{2\}andAfA\_\{f\}\. Architecture\-level injection therefore helps under data scarcity, and this reframes the central question of the paper: if architecture\-level injection improves performance, can the same knowledge be supplied through simulation instead, and how do the two channels relate?

![Refer to caption](https://arxiv.org/html/2607.20539v1/figures/F3_sim_ablation.png)Figure 2:Effect of simulation methodology and injection mode on testR2R^\{2\}, across three architectures spanning the prior\-intensity spectrum \(MLP, Hybrid\-NeuralODE, BioStruct\-ODE\+Cardinal\)\. The top row sweeps pre\-training sim ratio\|sim\|/\|real\|∈\{1,3,5,10,30\}\|\\mathrm\{sim\}\|/\|\\mathrm\{real\}\|\\in\\\{1,3,5,10,30\\\}, and the bottom row sweeps joint\-training sim mix weight∈\{0\.5,1,2,5\}\\in\\\{0\.5,1,2,5\\\}\. Each line depicts a sim variant \(composite\-biokinetic broad, single\-ODE narrow, random\-GP control\); dashed horizontal lines depict the corresponding no\-sim baselines that match those in Table[1](https://arxiv.org/html/2607.20539#S4.T1)\. Values are averaged over 5 seeds and theyy\-axis is per\-column\.
### 4\.2RQ2: Simulation prior injection

Simulation as a second injection routeThe same ODEs that we embed in the decoder for BioStruct\-ODE can also be sampled and integrated to produce a synthetic dataset of curves, which then pre\-trains or jointly trains an otherwise generic decoder\. We therefore ask two coupled questions:*how*should the synthetic dataset be constructed, and*how*should it be injected into training? We select three architectures spanning the prior\-intensity spectrum \(MLP, Hybrid\-NeuralODE, BioStruct\-ODE\+Cardinal\) and compare two injection modes: \(i\) pre\-training at five sim ratios and \(ii\) joint training at four mix weights\. Each setting is evaluated with three sim variants: composite\-biokinetic broad sampling, single\-ODE narrow sampling, and a non\-biokinetic random\-GP control that removes biokinetic structure while preserving curve smoothness\. Results are reported in Figure[2](https://arxiv.org/html/2607.20539#S4.F2)\.

Biokinetic specificity is the key factorInspecting Figure[2](https://arxiv.org/html/2607.20539#S4.F2)column by column, every architecture exhibits the same ordering: composite\-biokinetic broad sampling outperforms single\-ODE narrow sampling, and both outperform the random\-GP control by a margin that widens with the simulation budget\. Smoothness and curve\-shape regularity therefore cannot explain the gain; the simulation must carry biokinetic structure \(saturation, lag, carrying capacity\) for the prior to transfer\.

Pre\-training improves MLP the mostThe MLP column of Figure[2](https://arxiv.org/html/2607.20539#S4.F2)exhibits the largest improvement: composite\-broad pre\-training improvesR2R^\{2\}from the no\-sim baseline of0\.1680\.168to≈0\.515\\approx 0\.515at saturation, on par with a BioStruct\-ODE\+Cardinal trained from scratch on real data alone \(0\.5540\.554on its seven\-dataset subset\)\. Joint training, by contrast, peaks at low mix weights and then degrades; for an MLP the joint maximum isR2≈0\.30R^\{2\}\\approx 0\.30at mix weight11before declining toward the no\-sim baseline at higher weights, whereas pre\-training saturates and remains stable\.

A practical recipe for the data\-scarce settingThe match between MLP\+\+pre\-train \(R2≈0\.515R^\{2\}\\approx 0\.515\) and BioStruct\-ODE\+Cardinal\+\+nosim \(0\.5540\.554\) is our key substitution finding: the architecture and data channels behave as substitutes for the same biokinetic prior, and the gap between them lies within the seed\-level standard deviations reported in Table[1](https://arxiv.org/html/2607.20539#S4.T1)\. We therefore recommend composite\-biokinetic broad pre\-training at moderate sim ratios as a practical, data\-efficient strategy for practitioners with a generic decoder and limited real data\.

![Refer to caption](https://arxiv.org/html/2607.20539v1/figures/F4_ctx_protocol.png)Figure 3:Illustration of the test\-time context\-conditioning protocol used in RQ3 \(schematic, not measured\)\. A fixed window of timepoints \(orange band\) is held out as the evaluation set across all context densities, while context observations \(filled circles\) are drawn only from the complementary context pool \(blue band, depicted only when context is supplied\)\.
### 4\.3RQ3: Context conditioning

Does context replace priors?To make different context densities directly comparable, we split each test curve into a fixed back\-70%70\\%evaluation window and a front\-30%30\\%context pool\. Every baseline is then evaluated at three context densities \(0,1010, and30%30\\%of the curve\), with the back\-70%70\\%window held fixed across all densities so that only the inputs received during inference change\. The protocol is depicted in Figure[3](https://arxiv.org/html/2607.20539#S4.F3)and the results are reported in Table[2](https://arxiv.org/html/2607.20539#S4.T2)\.

Context helps; pre\-training transfers cleanlyAcross all eight trainable baselines in Table[2](https://arxiv.org/html/2607.20539#S4.T2), moving from ctx\-0%0\\%to ctx\-10%10\\%yields a small but consistent gain \(meanΔ​R2≈\+0\.012\\Delta R^\{2\}\\approx\+0\.012, meanΔ​Af≈−0\.010\\Delta A\_\{f\}\\approx\-0\.010\), and the additional gain from ctx\-10%10\\%to ctx\-30%30\\%is smaller still \(Δ​R2≈\+0\.005\\Delta R^\{2\}\\approx\+0\.005\)\. Every\+\+Pretrain sub\-row outperforms its scratch counterpart at every context density, with the largest absolute gain on the weakest backbone \(MLP\+\+Pretrain at ctx\-0%0\\%improvesR2R^\{2\}by about\+0\.347\+0\.347over MLP scratch\) and a much smaller gain on the strongest one \(BioStruct\-ODE\+Cardinal\+\+Pretrain at ctx\-0%0\\%improves it by about\+0\.009\+0\.009\)\.

Architecture, simulation, and context stack additivelyThe strongest single configuration is BioStruct\-ODE\+Cardinal\+\+Pretrain at ctx\-30%30\\%, attainingR2=0\.577R^\{2\}=0\.577, which improves on the strongest scratch configuration at the same context density \(0\.5640\.564\) and on the no\-context Cardinal\+Pretrain \(0\.5630\.563\)\. In practice,≈10%\\approx 10\\%of the curve is the operating point with the best gain\-per\-cost trade\-off, while simulation pre\-training and an architectural prior should still be applied on top of context\.

Table 2:Test\-timeR2R^\{2\}andAfA\_\{f\}at three context densities \(RQ3\)\. The\+\+Pretrainsub\-rows show the best simulation configuration \(composite\-biokinetic broad pre\-train, sim\_ratio=10=10\) for the three representative architectures\. The ctx\-0%0\\%column for scratch rows matches Table[1](https://arxiv.org/html/2607.20539#S4.T1)exactly\. Values are mean±std\{\}\_\{\\pm\\text\{std\}\}across55seeds, and the best value in each column is shown in bold\.R2↑R^\{2\}\\uparrowAf↓A\_\{f\}\\downarrowModel0%0\\%10%10\\%30%30\\%0%0\\%10%10\\%30%30\\%Mean0\.157±\.3360\.157\_\{\\pm\.336\}0\.157±\.3360\.157\_\{\\pm\.336\}0\.157±\.3360\.157\_\{\\pm\.336\}1\.418±\.2531\.418\_\{\\pm\.253\}1\.418±\.2531\.418\_\{\\pm\.253\}1\.418±\.2531\.418\_\{\\pm\.253\}ODE\-Fit0\.204±\.3420\.204\_\{\\pm\.342\}0\.215±\.3300\.215\_\{\\pm\.330\}0\.220±\.3250\.220\_\{\\pm\.325\}1\.338±\.1361\.338\_\{\\pm\.136\}1\.328±\.1341\.328\_\{\\pm\.134\}1\.322±\.1321\.322\_\{\\pm\.132\}MLP0\.168±1\.200\.168\_\{\\pm 1\.20\}0\.180±1\.100\.180\_\{\\pm 1\.10\}0\.185±1\.050\.185\_\{\\pm 1\.05\}1\.329±\.1491\.329\_\{\\pm\.149\}1\.318±\.1451\.318\_\{\\pm\.145\}1\.312±\.1421\.312\_\{\\pm\.142\}\+\+Pretrain0\.515±\.2050\.515\_\{\\pm\.205\}0\.528±\.1980\.528\_\{\\pm\.198\}0\.534±\.1930\.534\_\{\\pm\.193\}1\.272±\.1421\.272\_\{\\pm\.142\}1\.262±\.1391\.262\_\{\\pm\.139\}1\.256±\.1371\.256\_\{\\pm\.137\}GRU0\.494±\.2460\.494\_\{\\pm\.246\}0\.508±\.2380\.508\_\{\\pm\.238\}0\.514±\.2320\.514\_\{\\pm\.232\}1\.288±\.1531\.288\_\{\\pm\.153\}1\.278±\.1501\.278\_\{\\pm\.150\}1\.273±\.1481\.273\_\{\\pm\.148\}PINN0\.503±\.1880\.503\_\{\\pm\.188\}0\.515±\.1830\.515\_\{\\pm\.183\}0\.520±\.1800\.520\_\{\\pm\.180\}1\.287±\.1351\.287\_\{\\pm\.135\}1\.276±\.1321\.276\_\{\\pm\.132\}1\.270±\.1301\.270\_\{\\pm\.130\}ODE\-Guide0\.090±\.3460\.090\_\{\\pm\.346\}0\.100±\.3400\.100\_\{\\pm\.340\}0\.105±\.3360\.105\_\{\\pm\.336\}1\.437±\.2051\.437\_\{\\pm\.205\}1\.428±\.2021\.428\_\{\\pm\.202\}1\.423±\.2001\.423\_\{\\pm\.200\}Hybrid\-NeuralODE0\.342±\.3490\.342\_\{\\pm\.349\}0\.355±\.3400\.355\_\{\\pm\.340\}0\.361±\.3340\.361\_\{\\pm\.334\}1\.336±\.1391\.336\_\{\\pm\.139\}1\.325±\.1361\.325\_\{\\pm\.136\}1\.319±\.1341\.319\_\{\\pm\.134\}\+\+Pretrain0\.376±\.3100\.376\_\{\\pm\.310\}0\.382±\.3030\.382\_\{\\pm\.303\}0\.385±\.2980\.385\_\{\\pm\.298\}1\.315±\.1351\.315\_\{\\pm\.135\}1\.305±\.1331\.305\_\{\\pm\.133\}1\.300±\.1311\.300\_\{\\pm\.131\}BioStruct\-ODE \(Ours\)0\.521±\.2540\.521\_\{\\pm\.254\}0\.534±\.2470\.534\_\{\\pm\.247\}0\.540±\.2430\.540\_\{\\pm\.243\}1\.266±\.1361\.266\_\{\\pm\.136\}1\.256±\.1331\.256\_\{\\pm\.133\}1\.250±\.1311\.250\_\{\\pm\.131\}\+\+Cardinal0\.554±\.2120\.554\_\{\\pm\.212\}0\.561±\.2070\.561\_\{\\pm\.207\}0\.564±\.2030\.564\_\{\\pm\.203\}1\.235±\.0791\.235\_\{\\pm\.079\}1\.230±\.0771\.230\_\{\\pm\.077\}1\.227±\.0761\.227\_\{\\pm\.076\}\+\+Pretrain0\.563±\.205\\mathbf\{0\.563\}\_\{\\pm\.205\}0\.572±\.200\\mathbf\{0\.572\}\_\{\\pm\.200\}0\.577±\.196\\mathbf\{0\.577\}\_\{\\pm\.196\}1\.228±\.077\\mathbf\{1\.228\}\_\{\\pm\.077\}1\.221±\.075\\mathbf\{1\.221\}\_\{\\pm\.075\}1\.218±\.074\\mathbf\{1\.218\}\_\{\\pm\.074\}
### 4\.4RQ4: Failure\-mode analysis

What kind of error does pre\-training correct?To investigate this further, we evaluate the strongest configuration \(BioStruct\-ODE\-Cardinal\+\+Pretrain\) on two real test curves and contrast its predictions with the same model trained from scratch \(Figure[4](https://arxiv.org/html/2607.20539#S4.F4)\)\.

Lag and carrying capacity recoveredOn a low\-temperatureListerialag\-phase example \(Figure[4](https://arxiv.org/html/2607.20539#S4.F4)a\), the scratch model predicts immediate growth from time zero and overshoots the early measurements with an almost monotonic curve\.\+\+Pretrain, by contrast, remains near zero growth through the first few hours and only then begins to increase, tracking the ground truth\. On aYersiniastationary\-phase example \(Figure[4](https://arxiv.org/html/2607.20539#S4.F4)b\), the scratch model overshoots the plateau and continues to climb above the carrying capacity, while\+\+Pretrain approaches the correctKKand remains at the plateau\.

Errors map onto specific ODE termsThe lag correction corresponds to the lag dynamicsqqin the BioStruct\-ODE system, and the plateau correction corresponds to the carrying capacityKK; a similar argument extends in principle to the death termkdk\_\{d\}in the late phase of the curve\. A generic smoothing prior would not isolate these particular ODE terms but would instead shrink the prediction toward an arbitrary mean trajectory\.

![Refer to caption](https://arxiv.org/html/2607.20539v1/figures/F5_failure_modes.png)Figure 4:Failure\-mode comparison of the scratch and\+\+Pretrain predictions of BioStruct\-ODE\+Cardinal on two hand\-selected real test curves\.\(a\)Lag phase, drawn fromListeriaBuchanan19891989at5∘5\\,^\{\\circ\}C: the scratch model predicts immediate growth, while\+\+Pretrain captures the lag\.\(b\)Stationary phase, drawn from ComBaseYersiniaat0∘0\\,^\{\\circ\}C: the scratch model overshoots the plateau, while\+\+Pretrain converges near the correct level \(Section[4\.4](https://arxiv.org/html/2607.20539#S4.SS4)\)\.

## 5Conclusion

Biokinetic\-ODE\-derived simulation curves can substitute for a substantial portion of scarce real bioprocess data\. A unified comparison demonstrates that simulation pre\-training and architecture\-level priors act as substitutable channels, with biokinetic specificity as the key factor\. In particular, a generic neural decoder with simulation pre\-training matches a fully bio\-structured decoder, while random\-curve simulation fails and pre\-training outperforms joint training\.

#### Limitations

Cross\-organism transfer to unseen species remains unsolved \(Appendix[E\.2](https://arxiv.org/html/2607.20539#A5.SS2)\), and our scope excludes mammalian cell systems where industrial data privacy dominates\. Our evaluation is limited to bacterial batch cultivation; fed\-batch dynamics, perfusion, and continuous culture are not addressed\. We use55seeds per configuration, which limits the precision of variance estimates for borderline comparisons\. The RQ4 mechanism evidence is illustrative rather than aggregate, and the average\-case quantitative evidence is concentrated in Tables[1](https://arxiv.org/html/2607.20539#S4.T1)–[2](https://arxiv.org/html/2607.20539#S4.T2)and Figure[2](https://arxiv.org/html/2607.20539#S4.F2)\.

#### Future work

A natural next step is to extend the prediction interface into closed\-loop deployment with dynamic process control or online monitoring systems\. The substitutability finding also suggests hybrid pipelines that allocate effort between architecture engineering and simulation generation according to the available compute and engineering cost\.

## 6Dataset availability

All datasets used in this work are publicly available\.

## Institutional Review Board \(IRB\)

This research does not require IRB approval\.

## Acknowledgments

This work was supported by Institute of Information & communications Technology Planning & Evaluation \(IITP\) grant funded by the Korea government\(MSIT\) \(RS\-2024\-00346798,

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## Appendix ADataset Details

### A\.1Naming convention

We adopt paper\-facing dataset names throughout the paper, mapping each internal dataset identifier to the original source\. Body text uses the full name on first mention with an organism qualifier \(e\.g\.,Katipoglu\-Yazan 2023\(E\. coliNCM3722\)\) and the full name thereafter\. Tables and figures may use the short form when space requires it; the short form is then defined in the caption\.

Table 3:Dataset naming convention\. Body text uses the full name on first mention with an organism qualifier, and the full name thereafter\. Tables and figures may use the short form when space requires; the short form is then defined in the caption\.
### A\.2Dataset statistics

Table[4](https://arxiv.org/html/2607.20539#A1.T4)summarizes the 11 datasets used in the paper\.N rawis the original curve count from the source repository;N preprocis the remaining curve count after the per\-curvewith\_growthfilter, which retains a curve only if its single\-curve Monod\-Baranyi nonlinear least\-squares fit achievesR2≥0\.4R^\{2\}\\geq 0\.4\. Time horizon and time interval are means over the preprocessed set; the env axis lists the within\-dataset varying conditions\.Avgμmax\\mu\_\{\\max\}is the per\-curve fitted growth rate averaged across preprocessed curves and is used only as a coarse cross\-organism comparator; the model never sees these literature fits during training\.

Three datasets on which no baseline could reach a best meanR2≥0\.42R^\{2\}\\geq 0\.42at size L \(Yeast Y1000\+,S\. aureusBuchanan,S\. aureusComBase\) are excluded at the dataset level\.BL21 \(UCL\)reaches a mid\-tier bestR2≈0\.346R^\{2\}\\approx 0\.346but is retained for its own\-data value and for organism diversity\.

Table 4:Dataset statistics for the 11 datasets used in the paper\.N raw= original curve count from the source repository;N preproc= remaining curves after the per\-curvewith\_growthfilter \(single\-curve Monod\-Baranyi NLS fitR2≥0\.4R^\{2\}\\geq 0\.4retained\)\. Time horizon and time interval are means over the preprocessed set; the env axis lists within\-dataset varying conditions\.
### A\.3Train, validation, and test split

We use curve\-level random splits in which Test \(15%\) and Val \(15%\) are size\-invariant and only Train varies across three size variants S/M/L \(20% / 40% / 70% of total preprocessed curves\)\. Splits are seeded by\(organism\_id, seed\_idx\)forseed\_idx∈\{0,1,2,3,4\}\\texttt\{seed\\\_idx\}\\in\\\{0,1,2,3,4\\\}so that the five\-seed average is computed over genuinely distinct partitions\. Within Test, we additionally hold out the back 70% of timepoints per curve as the never\-seen\-as\-context evaluation set \(Section[4\.3](https://arxiv.org/html/2607.20539#S4.SS3)in the main paper\); all context densities therefore evaluate on identical timepoints\. The paper’s main results use size L only, so the training proportion is fixed at 70% and the full sweep is11×9×5=49511\\times 9\\times 5=495cells per\(sim\_mode, ctx\_ratio\)combination\.

Table 5:Per\-dataset train / validation / test split\. Test \(15%\) and Val \(15%\) are size\-invariant; the training set varies across three size variants S/M/L \(20% / 40% / 70% of total preprocessed curves\)\. Splits are at the curve level with random sampling, seeded by\(organism\_id, seed\_idx\)forseed\_idx∈\{0,1,2,3,4\}\\in\\\{0,1,2,3,4\\\}\. The paper’s main results use size L only\.

## Appendix BSimulation Generation Protocol

### B\.1Simulation variants

This section defines the simulation curve synthesis protocol used in the RQ2 ablation\. Following the three factors introduced in Section[3\.3](https://arxiv.org/html/2607.20539#S3.SS3)of the main paper, each variant is fully specified by \(Factor A\) the ODE family that is integrated, \(Factor B\) the parameter distribution that is sampled, and \(Factor C\) the resulting position on the biokinetic specificity scale \(random\-GP control / single\-ODE narrow / composite\-ODE broad\)\.

#### ODE families\.

We consider four biokinetic ODE families and one non\-biokinetic control\. Throughout,N​\(t\)N\(t\)is the cell density,S​\(t\)S\(t\)the substrate concentration, andq​\(t\)q\(t\)a lag\-phase adaptation state\.

Monod\(Monod,[1949](https://arxiv.org/html/2607.20539#bib.bib1)\)models substrate\-limited growth as

d​Nd​t\\displaystyle\\frac\{dN\}\{dt\}=μ​\(S\)​N,\\displaystyle\\;=\\;\\mu\(S\)\\,N,μ​\(S\)\\displaystyle\\mu\(S\)=μmax​SKs\+S,\\displaystyle\\;=\\;\\mu\_\{\\max\}\\,\\tfrac\{S\}\{K\_\{s\}\+S\},d​Sd​t\\displaystyle\\tfrac\{dS\}\{dt\}=−μ​\(S\)​NYx​s\.\\displaystyle\\;=\\;\-\\,\\tfrac\{\\mu\(S\)\\,N\}\{Y\_\{xs\}\}\.\(10\)
Logisticignores substrate dynamics and uses a carrying\-capacity term:

d​Nd​t=μmax​N​\(1−NK\)\.\\frac\{dN\}\{dt\}\\;=\\;\\mu\_\{\\max\}\\,N\\\!\\left\(1\-\\tfrac\{N\}\{K\}\\right\)\.\(11\)
Gompertz\(Zwieteringet al\.,[1990](https://arxiv.org/html/2607.20539#bib.bib4)\)is a sigmoidal closed form

N​\(t\)=N0\+A​e−exp⁡\(μmax​eA​\(λ−t\)\+1\),N\(t\)\\;=\\;N\_\{0\}\+A\\,e^\{\-\\exp\\\!\\left\(\\tfrac\{\\mu\_\{\\max\}e\}\{A\}\(\\lambda\-t\)\+1\\right\)\},\(12\)withA=K−N0A=K\-N\_\{0\}, lag timeλ\\lambda, ande=exp⁡\(1\)e=\\exp\(1\)\.

Baranyi & Roberts\(Baranyi and Roberts,[1994](https://arxiv.org/html/2607.20539#bib.bib2)\)introduces a two\-state ODE with explicit lag dynamics:

d​Nd​t\\displaystyle\\frac\{dN\}\{dt\}=μmax​q1\+q​\(1−NK\)​N,\\displaystyle\\;=\\;\\mu\_\{\\max\}\\,\\frac\{q\}\{1\+q\}\\,\\\!\\left\(1\-\\tfrac\{N\}\{K\}\\right\)\\\!N,d​qd​t\\displaystyle\\frac\{dq\}\{dt\}=μmax​q\.\\displaystyle\\;=\\;\\mu\_\{\\max\}\\,q\.\(13\)
Rosso composite\(Rossoet al\.,[1995](https://arxiv.org/html/2607.20539#bib.bib6)\)couples the Monod substrate term with cardinal envelopes for temperature and pH:

μ​\(S,T,pH\)=γ​\(T\)​γ​\(pH\)​μmax​SKs\+S,\\mu\(S,T,\\mathrm\{pH\}\)\\;=\\;\\gamma\(T\)\\,\\gamma\(\\mathrm\{pH\}\)\\,\\mu\_\{\\max\}\\,\\tfrac\{S\}\{K\_\{s\}\+S\},\(14\)where eachγ\\gammais the Rosso cardinal function with three reference points \(e\.g\.,Tmin,Topt,TmaxT\_\{\\min\},T\_\{\\mathrm\{opt\}\},T\_\{\\max\}\) and is positive only inside the biological envelope\.

#### Parameter sampling\.

Each simulated curve draws parameters from a distribution centered at the organism’s literature warm\-start mean \(the same values used for the BioStruct\-ODE ParamHead bias initialization, Appendix[C](https://arxiv.org/html/2607.20539#A3)\)±\\pma sampling spread\. Thebroad uniformregime uses literature±50%\\pm 50\\%, while theliterature\-tightregime uses literature±10%\\pm 10\\%\. The initial conditionN0N\_\{0\}is sampled from\[0\.01,0\.05\]\[0\.01,0\.05\]in normalized units, and we add a multiplicative log\-normal observation noise \(σ=0\.05\\sigma=0\.05\)\.

#### Random\-GP control\.

Thesim\_random\_gpvariant uses no biokinetic ODE at all\. It draws length\-scalesℓ∼Uniform​\[2,12\]\\ell\\sim\\mathrm\{Uniform\}\[2,12\]h and samples from a zero\-mean Gaussian process with an RBF kernel of scaleℓ\\ell, with no monotonicity, saturation, or other biological structure imposed\. This control isolates whether the benefit of simulation pre\-training is due to*biokinetic specificity*per se or merely to exposing the model to many smooth curves\.

#### Variant table\.

The four sim variants used in our experiments are summarized in Table[6](https://arxiv.org/html/2607.20539#A2.T6)\.

Table 6:Simulation dataset variants used in RQ2\. Per\-organism sampling centers each parameter at the literature warm\-start mean±\\pmthe indicated sampling spread;sim\_random\_gpis a non\-biokinetic control \(Gaussian Process samples\)\. The fourth row,sim\_per\_family, is used for the RQ2 ODE\-family ablation with one ODE family per variant\.Parameter ranges \(sim\_composite\_broad / sim\_single\_narrow\):μmax∈\[0\.05,1\.50\]/\[0\.40,0\.95\]\\mu\_\{\\max\}\\in\[0\.05,1\.50\]/\[0\.40,0\.95\]h\-1;K∈\[0\.10,1\.50\]/\[0\.50,1\.00\]K\\in\[0\.10,1\.50\]/\[0\.50,1\.00\]\(normalized\); lag∈\[0,8\]/\[0,4\]\\in\[0,8\]/\[0,4\]h\.

## Appendix CBaseline Architecture and Implementation

### C\.1Architecture and parameter counts

Table[7](https://arxiv.org/html/2607.20539#A3.T7)lists the encoder, decoder, hidden / latent dimensions, and parameter counts for all nine baselines at three size variants\. All neural baselines share the same encoder stack \(EnvEncoderfor environmental conditions andContextEncoderfor partial\-observation context\) and differ only in the decoder head, which isolates the architectural\-prior contribution from encoder capacity\. ThePINNbaseline reuses the MLP decoder verbatim and adds an autograd\-based physics\-residual auxiliary loss term during training \(defined in Appendix[D\.1](https://arxiv.org/html/2607.20539#A4.SS1)\)\. TheODE\-Guide,Hybrid\-NeuralODE, and BioStruct\-ODE family progressively integrate a biokinetic ODE into the forward pass with increasing structure \(closed\-form template→\\rightarrowtwo\-state ODE with neural correction→\\rightarrowthree\-state Monod\-Baranyi with ParamHead and Rosso cardinal env\-conditioning\)\. TheMeanandODE\-Fitbaselines are non\-parametric and have no learned weights\.

TheODE\-Fitbaseline fits a Gompertz ODE per training curve with nonlinear least squares; the per\-curve parameters\(μ,K,λ,N0\)\(\\mu,K,\\lambda,N\_\{0\}\)are averaged across the dataset’s training set, and the averaged parameters are applied to every test curve of the same dataset\. This protocol uses no environmental conditioning, paralleling theMeanbaseline but with biokinetic structure \(Appendix[D\.1](https://arxiv.org/html/2607.20539#A4.SS1)defines theL4L\_\{4\}Huber regression loss used by all neural baselines\)\. The BioStruct\-ODE familyParamHeadbiases are warm\-started from organism\-specific literature Monod fits\.

Table 7:Per\-baseline architecture and parameter counts at three size variants \(S / M / L\)\. All neural baselines share the same encoder stack \(EnvEncoder \+ ContextEncoder\) and differ only in the decoder head; this isolates the architectural\-prior contribution from encoder capacity\. Mean and ODE\-Fit are non\-parametric baselines without learned weights\.shared= EnvEncoder \+ ContextEncoder; Hidden is the per\-size hidden dim\.

## Appendix DTraining Details

### D\.1Loss functions

All neural baselines are trained against a regression loss on the observed timepoints\. We consider five loss families and adoptL4L\_\{4\}\(Huber\) as the paper\-wide default based on a preliminary loss ablation; letΩ=\{\(i,t\)\}\\Omega=\\\{\(i,t\)\\\}denote the set of \(curve, timepoint\) observations\. The MSE and MAE losses are

ℒL​0\\displaystyle\\mathcal\{L\}\_\{L0\}=1\|Ω\|​∑\(i,t\)∈Ω\(y^i​\(t\)−yi​\(t\)\)2,\\displaystyle\\;=\\;\\tfrac\{1\}\{\|\\Omega\|\}\\\!\\\!\\sum\_\{\(i,t\)\\in\\Omega\}\\\!\\bigl\(\\hat\{y\}\_\{i\}\(t\)\-y\_\{i\}\(t\)\\bigr\)^\{2\},\(15\)ℒL​1\\displaystyle\\mathcal\{L\}\_\{L1\}=1\|Ω\|​∑\(i,t\)∈Ω\|y^i​\(t\)−yi​\(t\)\|\.\\displaystyle\\;=\\;\\tfrac\{1\}\{\|\\Omega\|\}\\\!\\\!\\sum\_\{\(i,t\)\\in\\Omega\}\\\!\\bigl\|\\hat\{y\}\_\{i\}\(t\)\-y\_\{i\}\(t\)\\bigr\|\.\(16\)The time\-consistency and curvature regularizers add a derivative penalty to the MSE base,

ℒL​2\\displaystyle\\mathcal\{L\}\_\{L2\}=ℒL​0\+λtc​∑\(i,t\)\|∂ty^i​\(t\)\|2,\\displaystyle\\;=\\;\\mathcal\{L\}\_\{L0\}\+\\lambda\_\{\\mathrm\{tc\}\}\\\!\\\!\\sum\_\{\(i,t\)\}\\bigl\|\\partial\_\{t\}\\hat\{y\}\_\{i\}\(t\)\\bigr\|^\{2\},\(17\)ℒL​3\\displaystyle\\mathcal\{L\}\_\{L3\}=ℒL​0\+λc​∑\(i,t\)\|∂t2y^i​\(t\)\|2\.\\displaystyle\\;=\\;\\mathcal\{L\}\_\{L0\}\+\\lambda\_\{\\mathrm\{c\}\}\\\!\\\!\\sum\_\{\(i,t\)\}\\bigl\|\\partial\_\{t\}^\{2\}\\hat\{y\}\_\{i\}\(t\)\\bigr\|^\{2\}\.\(18\)The default Huber loss with thresholdδ=0\.1\\delta=0\.1is

ℒL​4=1\|Ω\|​∑\(i,t\)∈Ωρδ​\(y^i​\(t\)−yi​\(t\)\)\.\\mathcal\{L\}\_\{L4\}\\;=\\;\\tfrac\{1\}\{\|\\Omega\|\}\\\!\\\!\\sum\_\{\(i,t\)\\in\\Omega\}\\\!\\rho\_\{\\delta\}\\\!\\bigl\(\\hat\{y\}\_\{i\}\(t\)\-y\_\{i\}\(t\)\\bigr\)\.\(19\)
ThePINNbaseline adds an auxiliary physics\-residual loss in the form of a logistic ODE,

ℒphys=1\|Tc\|​∑t∈Tc\(d​y^d​t−μ​y^​\(1−y^/K\)\)2,\\mathcal\{L\}\_\{\\mathrm\{phys\}\}\\;=\\;\\frac\{1\}\{\|T\_\{c\}\|\}\\sum\_\{t\\in T\_\{c\}\}\\left\(\\frac\{d\\hat\{y\}\}\{dt\}\\;\-\\;\\mu\\,\\hat\{y\}\\,\(1\-\\hat\{y\}/K\)\\right\)^\{2\},\(20\)where\(μ,K\)\(\\mu,K\)are taken from the per\-dataset Gompertz fits used by theODE\-Fitbaseline \(so the loss\-channel and template\-channel see*the same*kinetic prior, isolating the channel comparison\),TcT\_\{c\}is a random sample of collocation timepoints drawn each minibatch, andd​y^/d​td\\hat\{y\}/dtis computed by automatic differentiation\. The total PINN training loss isℒtotal=\(1−λ\)​ℒdata\+λ​ℒphys\\mathcal\{L\}\_\{\\mathrm\{total\}\}=\(1\-\\lambda\)\\,\\mathcal\{L\}\_\{\\mathrm\{data\}\}\+\\lambda\\,\\mathcal\{L\}\_\{\\mathrm\{phys\}\}withλ=0\.1\\lambda=0\.1\(no per\-datasetλ\\lambdatuning\)\.

### D\.2Optimizer, schedule, and computational budget

We use AdamW with learning rate3×10−43\\times 10^\{\-4\}, weight decay10−510^\{\-5\}, linear warmup over the first 10 epochs, and cosine decay thereafter\. Training is monitored on validationR2R^\{2\}atctx 0%with early stopping \(patience 5,Δmin=10−4\\Delta\_\{\\min\}=10^\{\-4\}\)\. All experiments run on8×8\\timesNVIDIA RTX 3090 GPUs at five concurrent workers per GPU \(40 workers total\); the full RQ1 sweep \(9​baselines×11​datasets×5​seeds×size L9\\text\{ baselines\}\\times 11\\text\{ datasets\}\\times 5\\text\{ seeds\}\\times\\text\{size~L\}\) completes in approximately one wall\-clock hour\. Aggregate compute budgets are RQ1≈16\\approx 16GPU\-hours, RQ2≈80\\approx 80GPU\-hours, RQ3≈24\\approx 24GPU\-hours, and RQ4 \(post\-hoc analysis only\)≈0\\approx 0, for a total of about120120GPU\-hours of paper\-supporting compute\.

## Appendix EAblation Studies

### E\.1Biokinetic component ablation

Table[8](https://arxiv.org/html/2607.20539#A5.T8)reports the contribution of each binary biokinetic component \(warm\-start, death term, template choice\) on the two main bio\-structured models\. Each component is toggled while the other two are held at the paper default \(Warm\-start on, Death term on, Template = Monod\-Baranyi\); the default row matches Table[1](https://arxiv.org/html/2607.20539#S4.T1)exactly for both models\.

Table 8:Biokinetic component ablation on the two bio\-structured models \(Ours\)\. Each component is toggled binary while keeping the other two at the paper default \(Warm\-start on, Death term on, Template = Monod–Baranyi\); the default row matches Table[1](https://arxiv.org/html/2607.20539#S4.T1)exactly\. All three components contribute small but consistent gains, and the death term yields the largest singleAfA\_\{f\}improvement\.The death term contributes the largest single Af improvement \(1\.330→\\rightarrow1\.266 for BioStruct\-ODE; 1\.290→\\rightarrow1\.235 for BioStruct\-ODE\+Cardinal\), with a smallerR2R^\{2\}lift of\+0\.011\+0\.011to\+0\.012\+0\.012; its primary effect is on stationary\-phase fidelity\. Warm\-start gives the smallestR2R^\{2\}gain \(\+0\.006\+0\.006to\+0\.008\+0\.008\) but a consistentAfA\_\{f\}improvement of about−0\.02\-0\.02to−0\.03\-0\.03, stabilizing the kinetic head without dramatic effect\. Monod\-Baranyi outperforms Gompertz by a stable margin \(≈\+0\.02\\approx\+0\.02R2R^\{2\},≈−0\.03\\approx\-0\.03AfA\_\{f\}\) across both models, since the substrate\-dynamics term in Monod\-Baranyi provides a richer prior than the logistic\-only Gompertz form\.

### E\.2Cross\-organism transfer

We evaluate cross\-organism transfer with BioStruct\-ODE \(Ours, scratch\) by training on each organism’s training set and evaluating on every other organism’s test set, producing the11×1111\\times 11matrix in Table[9](https://arxiv.org/html/2607.20539#A5.T9)\. The bold diagonal corresponds to the within\-organism baseline and matches the per\-dataset BioStruct\-ODE means in Table[1](https://arxiv.org/html/2607.20539#S4.T1)\.

Table 9:Cross\-organism transfer matrix for BioStruct\-ODE \(Ours, scratch\)\. Cell value=R2=R^\{2\}when training on Source organism’s training set and evaluating on Target organism’s test set\.Bold diagonal= within\-organism baseline \(matches the per\-dataset BioStruct\-ODE values int1\_per\_cell\.csv\)\. Off\-diagonal mean \(0\.108\) is far below the diagonal mean \(0\.572\), giving aΔ=−0\.46\\Delta=\-0\.46that justifies the per\-organism scope of the main paper\. Short forms: BL21 = BL21 \(UCL\); Katipoglu23 = Katipoglu\-Yazan 2023; Faure23 = Faure 2023; Buchan89 = Buchanan 1989; Zaika94 = Zaika 1994; Eifert97 = Eifert 1997; CB\-Ec/Lm/Yer/Sal/Pse = ComBase \(E\. coli / Listeria / Yersinia / Salmonella / Pseudomonads\)\.The off\-diagonal mean \(0\.108\) is far below the diagonal mean \(0\.572\), giving a gap ofΔ=−0\.464\\Delta=\-0\.464\. Phylogenetically close organisms recover only 30–40% of the within\-organismR2R^\{2\}\(e\.g\., the fourE\. colivariants among themselves\), and transfer to non\-E\. coliorganisms is essentially uniform nearR2<0\.2R^\{2\}<0\.2\. This is the direct quantitative basis for adopting a per\-organism scope in the main paper\.