Biologically-Informed Neural Networks

• BINNs are neural models that embed biological structure into architecture, loss functions, and optimization to reduce hypothesis space while enhancing interpretability.
• They combine mechanistic ODE/PDE formulations, clinical pipelines, and pathway-guided designs to integrate domain-specific constraints in data-driven modeling.
• BINNs balance data fidelity with embedded biological priors, enabling reliable mechanistic recovery and improved predictions across clinical, genomic, and brain-inspired applications.

Biologically-informed Neural Networks (BINNs) are neural models whose architecture, loss, optimization, or internal variable structure is designed to reflect known biological, clinical, anatomical, or process-based organization rather than treating the task as an unstructured input–output mapping. In contemporary usage, the term does not denote a single canonical architecture. Instead, it covers several related research programs: mechanistic neural ODE/PDE models that preserve differential-equation structure while learning unknown constitutive laws; clinically structured pipelines that hard-wire tissue hierarchies; pathway- and process-guided predictors that constrain connectivity by curated biology; and brain-inspired networks whose geometry, plasticity, or excitation–inhibition organization is biologically motivated (Lagergren et al., 2020, Beers et al., 2017, Kontolati et al., 16 Oct 2025).

1. Conceptual scope and definitional variants

Across the literature, BINNs are unified less by a single implementation than by a common design principle: prior biological structure is moved from informal interpretation into the model itself. In mechanistic settings, this means preserving a reaction–diffusion or growth-law skeleton and learning only unknown operators or coefficients (Crossley et al., 8 Jul 2026). In clinical imaging, it means structuring prediction around known tissue containment or mutual exclusivity relations, such as whole tumor versus enhancing and non-enhancing subregions (Beers et al., 2017). In genomics and systems biology, it means masking connections by pathways, gene–omics mappings, or process models so that latent units correspond to biological entities rather than arbitrary hidden features (Pedersen et al., 2023).

Family	Core prior	Representative papers
Mechanistic BINNs	ODE/PDE structure, residual losses, biological constraints	(Lagergren et al., 2020, Crossley et al., 11 Jun 2026, Crossley et al., 8 Jul 2026, Lavery et al., 20 Apr 2026, Nardini, 2023)
Structured clinical/pathway BINNs	Tissue hierarchy, pathway masks, process modules	(Beers et al., 2017, Pedersen et al., 2023, Kontolati et al., 16 Oct 2025, Xu et al., 2 Feb 2025)
Brain-inspired BINNs	Spatial embedding, connectomics, E/I balance, plasticity rules	(Wołczyk et al., 2019, Kilgore et al., 2024, Laborieux et al., 2021, Prasanth et al., 20 Jan 2026, Dellaferrera et al., 2021, Liu et al., 2019)

This suggests that BINNs are best understood as an umbrella category for models that inject biological inductive bias at one or more of four levels: structure, inputs, learning objective, or optimization dynamics. A recurrent theme is that biological information is used to reduce the effective hypothesis space while preserving interpretability.

2. Mechanistic BINNs for differential-equation discovery

A major BINN lineage extends physics-informed neural networks by assuming that the governing equations are only partially known. In the reaction–diffusion formulation for wound-healing data, the state surrogate $u_{\mathrm{MLP}}(x,t)$ is coupled to neural approximations of density-dependent diffusivity and growth,

$$u_t = (D_{\mathrm{MLP}}(u)\,u_x)_x + G_{\mathrm{MLP}}(u)\,u,$$

with training driven by a data term, a PDE residual term computed by automatic differentiation, and biologically motivated constraint terms on ranges and monotonicity (Lagergren et al., 2020). On PC-3 scratch-assay data, inspecting the learned $D_{\mathrm{MLP}}(u)$, $G_{\mathrm{MLP}}(u)$, and a learned delay factor $T_{\mathrm{MLP}}(t)$ led to a compact delay reaction–diffusion model with hybrid FKPP/Porous-FKPP diffusion, logistic growth, and a logistic-in-time delay; that BINN-guided model achieved the lowest GLS error for all six datasets and the best AIC except in one case (Lagergren et al., 2020).

Subsequent work formalized mechanistic operator recovery for broader PDE families. For one-dimensional advection–diffusion–reaction systems,

$$\frac{\partial u}{\partial t} = \frac{\partial}{\partial x}\left(D(u)\frac{\partial u}{\partial x} - uV(u)\right) + uG(u),$$

the ADR–BINN uses one solution network $u_\theta(x,t)$ and separate MLPs for $D(u)$, $V(u)$, and $G(u)$, trained jointly by combining data misfit, PDE residual, and boundary-condition losses (Crossley et al., 8 Jul 2026). That study shows that reliable operator recovery depends on balanced data and PDE losses, moderate rather than maximal expressivity, intermediate learning rates, and intermediate batch sizes; it also emphasizes that a good learned surrogate $$u_t = (D_{\mathrm{MLP}}(u)\,u_x)_x + G_{\mathrm{MLP}}(u)\,u,$$0 does not by itself imply correct mechanism recovery, and that forward simulation with the recovered operators, $$u_t = (D_{\mathrm{MLP}}(u)\,u_x)_x + G_{\mathrm{MLP}}(u)\,u,$$1, is the more stringent diagnostic (Crossley et al., 8 Jul 2026).

Another extension adds explicit noise modeling. In population-growth BINNs, the latent trajectory $$u_t = (D_{\mathrm{MLP}}(u)\,u_x)_x + G_{\mathrm{MLP}}(u)\,u,$$2 and growth law $$u_t = (D_{\mathrm{MLP}}(u)\,u_x)_x + G_{\mathrm{MLP}}(u)\,u,$$3 are learned under the mechanistic constraint

$$u_t = (D_{\mathrm{MLP}}(u)\,u_x)_x + G_{\mathrm{MLP}}(u)\,u,$$4

while observations follow a heteroscedastic Gaussian model with

$$u_t = (D_{\mathrm{MLP}}(u)\,u_x)_x + G_{\mathrm{MLP}}(u)\,u,$$5

Training minimizes a negative log-likelihood plus ODE residual and positivity penalties, so that $$u_t = (D_{\mathrm{MLP}}(u)\,u_x)_x + G_{\mathrm{MLP}}(u)\,u,$$6 are learned jointly (Crossley et al., 11 Jun 2026). On synthetic logistic, Gompertz, and Richards data, the framework recovered the latent trajectory, crowding function, and noise profile across additive, intermediate, and multiplicative regimes; it also produced calibrated uncertainty bands, with approximately $$u_t = (D_{\mathrm{MLP}}(u)\,u_x)_x + G_{\mathrm{MLP}}(u)\,u,$$7 of data within $$u_t = (D_{\mathrm{MLP}}(u)\,u_x)_x + G_{\mathrm{MLP}}(u)\,u,$$8 and approximately $$u_t = (D_{\mathrm{MLP}}(u)\,u_x)_x + G_{\mathrm{MLP}}(u)\,u,$$9 within $D_{\mathrm{MLP}}(u)$0 (Crossley et al., 11 Jun 2026). This repositions noise from nuisance term to mechanistic object.

The mechanistic program has also been extended from $D_{\mathrm{MLP}}(u)$1 to $D_{\mathrm{MLP}}(u)$2 reaction–diffusion systems. In lung-cancer microscopy, BINNs were used to learn $D_{\mathrm{MLP}}(u)$3 and $D_{\mathrm{MLP}}(u)$4 from time-lapse cell-density fields and then pass the learned constitutive functions to symbolic regression, yielding closed-form replicate-specific PDEs for PC9 cell dynamics (Lavery et al., 20 Apr 2026). A related study used BINNs to learn effective one-compartment PDE surrogates for stochastic agent-based models of collective migration, showing that BINN-guided PDEs can forecast held-out times and, when combined with multivariate interpolation, predict behavior at previously unexplored parameter values even when the mean-field PDE is ill-posed or naturally two-compartment (Nardini, 2023).

3. Clinical and biomedical structure as architectural prior

In clinical imaging, BINNs can mean that the segmentation pipeline itself mirrors known tissue organization. A canonical example is glioma segmentation with sequential 3D U-Nets arranged as a tree. The first stage predicts whole tumor (WT), the second conditions enhancing tumor (ET) and tumor core (TC) prediction on the WT mask, and later refinement networks take MRI channels together with WT, ET, and TC labelmaps to learn tumor-specific cleanup rules (Beers et al., 2017). The hierarchy encodes the fact that edema surrounds tumor, that ET and non-enhancing/necrotic core are subregions of WT, and that those subregions are mutually exclusive by label definition even if no hard logical constraint is imposed in the loss (Beers et al., 2017).

Technically, all subnetworks use a 3D U-Net backbone on $D_{\mathrm{MLP}}(u)$5 patches, ReLU hidden activations, sigmoid binary outputs, batch normalization after each convolution, NAdam optimization, and a soft Dice loss. The WT stage is itself two-scale: a low-resolution model at $D_{\mathrm{MLP}}(u)$6 mm isotropic resolution is used to produce WT pseudo-probability maps, which are thresholded, upsampled, and appended as an extra input channel to a second high-resolution WT model. ET and TC models then use the WT labelmap as an additional input channel during training and inference (Beers et al., 2017).

On the BraTS 2017 validation set, this sequentially conditioned design achieved mean Dice scores of $D_{\mathrm{MLP}}(u)$7 for WT, $D_{\mathrm{MLP}}(u)$8 for ET, and $D_{\mathrm{MLP}}(u)$9 for TC, with leaderboard positions $G_{\mathrm{MLP}}(u)$0st, $G_{\mathrm{MLP}}(u)$1th, and $G_{\mathrm{MLP}}(u)$2th among $G_{\mathrm{MLP}}(u)$3 entrants (Beers et al., 2017). The significance is not only the scores but the design pattern: coarse biologically meaningful masks first, finer tissue compartment predictions later, and learned post-processing in place of generic morphology. At the same time, the paper is explicit that biological informativeness alone did not guarantee top leaderboard rank, and that ET and TC remain vulnerable to WT error propagation (Beers et al., 2017).

4. Pathway-, omics-, and process-guided BINNs

In high-dimensional genomics, BINNs often encode prior biology as sparse connectivity masks. In the prostate-cancer P-NET line, gene-level inputs connect only to pathway units permitted by Reactome membership and hierarchy, so most dense connections are masked out. A reusability study confirmed that this Reactome-guided sparsification is not merely parameter reduction: when the same sparsity level was preserved but the mask was randomly permuted, AUPR decreased, demonstrating that biologically correct topology mattered (Pedersen et al., 2023). The same study also showed that distinct biologically informed architectures can disagree systematically on individual patients even when overall metrics are similar, indicating architecture-specific sensitivity to different aspects of the data (Pedersen et al., 2023).

A more recent genomic-prediction formulation generalizes this idea by introducing masked pathway subnetworks between SNPs and phenotype. With genotype matrix $G_{\mathrm{MLP}}(u)$4, latent biological layers $G_{\mathrm{MLP}}(u)$5, and mask matrices $G_{\mathrm{MLP}}(u)$6, each entity at layer $G_{\mathrm{MLP}}(u)$7 is represented by a small MLP operating only on the subset of upstream features allowed by $G_{\mathrm{MLP}}(u)$8; a residual network handles unannotated SNPs, and an integrator network combines final latent variables and residual output into the phenotype prediction (Kontolati et al., 16 Oct 2025). In maize flowering-time prediction, this design used expression-based feature selection and eQTL mapping to build SNP→gene masks, but required genotype alone at inference time. Under sparse-data conditions, it improved rank-correlation accuracy by up to $G_{\mathrm{MLP}}(u)$9 within and across subpopulations, and in a synthetic metabolomics benchmark with complete domain knowledge it reduced prediction error by $T_{\mathrm{MLP}}(t)$0 relative to conventional neural nets (Kontolati et al., 16 Oct 2025). The same work introduced a correlation-based soft constraint,

$T_{\mathrm{MLP}}(t)$1

to align latent variables with measured intermediates without forcing exact reconstruction (Kontolati et al., 16 Oct 2025).

Process-guided BINNs extend the same logic to Earth-system biogeochemistry. In a Biogeochemistry-Informed Neural Network for soil organic carbon, a vectorized, differentiable implementation of the CLM5 soil-carbon module is embedded directly in the NN. The network maps $T_{\mathrm{MLP}}(t)$2 environmental covariates and location to $T_{\mathrm{MLP}}(t)$3 biogeochemical parameters constrained to literature-based ranges; the process core then computes the steady-state carbon profile analytically from the matrix-form model, and gradients backpropagate through CLM5 into the network (Xu et al., 2 Feb 2025). Trained on $T_{\mathrm{MLP}}(t)$4 observed SOC profiles across the conterminous United States, this BINN recovered process fields whose spatial patterns agreed with the PRODA benchmark with an average correlation coefficient of $T_{\mathrm{MLP}}(t)$5, achieved about $T_{\mathrm{MLP}}(t)$6 NSE on SOC prediction, and improved computational efficiency by more than $T_{\mathrm{MLP}}(t)$7 times over PRODA (Xu et al., 2 Feb 2025). Here the “biology” is biogeochemical process knowledge rather than cell-level physiology, but the formal principle is the same: unknown parameter fields are learned inside a mechanistic forward model rather than in a detached regression head.

5. Brain-inspired architectures, dynamics, and optimizers

A separate BINN tradition imports principles from neural tissue organization rather than differential equations. One line augments dense networks with explicit neuron positions in $T_{\mathrm{MLP}}(t)$8 and adds a transport cost,

$T_{\mathrm{MLP}}(t)$9

together with a density cost,

$$\frac{\partial u}{\partial t} = \frac{\partial}{\partial x}\left(D(u)\frac{\partial u}{\partial x} - uV(u)\right) + uG(u),$$0

so that strong long-range connections are penalized and neurons cannot collapse into a point (Wołczyk et al., 2019). On dual-task MNIST/Fashion-MNIST experiments, these spatial regularizers caused neurons to self-organize into task-specific clusters, and split subnetworks retained essentially full performance in concatenated and mixed settings, unlike regular networks (Wołczyk et al., 2019).

Another line constrains network dynamics and synapses more directly. In spiking neural networks, enforcing Dale’s law–like sign constraints and biologically realistic excitatory:inhibitory ratios led to reliable training at low firing rates and under noisy updates; networks with an $$\frac{\partial u}{\partial t} = \frac{\partial}{\partial x}\left(D(u)\frac{\partial u}{\partial x} - uV(u)\right) + uG(u),$$1 excitatory:inhibitory balance were particularly robust, and Van Rossum distances showed that inhibitory–inhibitory spike trains were more diverse than excitatory–excitatory ones in successful models (Kilgore et al., 2024). Connectomics has also been used as a hard architectural prior: BioNIC constructs biological layers whose widths match excitatory cell counts from a single mouse V1 cortical column, applies adjacency masks derived from MICrONS connectivity, and includes graded inhibition from inhibitory in-degree together with Hebbian plasticity, homeostatic regulation, LayerNorm, and synaptic noise. On FER-2013 it reached $$\frac{\partial u}{\partial t} = \frac{\partial}{\partial x}\left(D(u)\frac{\partial u}{\partial x} - uV(u)\right) + uG(u),$$2 accuracy, comparable to a conventional baseline at $$\frac{\partial u}{\partial t} = \frac{\partial}{\partial x}\left(D(u)\frac{\partial u}{\partial x} - uV(u)\right) + uG(u),$$3 (Prasanth et al., 20 Jan 2026).

Learning rules themselves can be biologically informed. In binarized neural networks, metaplasticity has been implemented by treating the hidden real-valued weight $$\frac{\partial u}{\partial t} = \frac{\partial}{\partial x}\left(D(u)\frac{\partial u}{\partial x} - uV(u)\right) + uG(u),$$4 as a synaptic internal state and scaling reinforcing updates by $$\frac{\partial u}{\partial t} = \frac{\partial}{\partial x}\left(D(u)\frac{\partial u}{\partial x} - uV(u)\right) + uG(u),$$5, thereby decreasing plasticity with metaplastic depth; this reduced catastrophic forgetting and performed almost as well as EWC on permuted MNIST while also helping in stream-learning settings without task boundaries (Laborieux et al., 2021). In CHL and EP networks, adding neuronal adaptation through

$$\frac{\partial u}{\partial t} = \frac{\partial}{\partial x}\left(D(u)\frac{\partial u}{\partial x} - uV(u)\right) + uG(u),$$6

improved MNIST and CIFAR-10 performance and made local updates more closely aligned with backpropagation directions (Kubo et al., 2022). GRAPES, a biologically inspired optimizer, modulates each neuron’s error signal by a factor derived from the distribution of synaptic weights around that node, improving convergence and classification in ANNs and SNNs while reducing required optimizer state (Dellaferrera et al., 2021). At the dynamical extreme, McCulloch–Pitts networks with asymmetric connections and local maximum-likelihood updates can store periodic attractors rather than static patterns, and the expected weight changes reproduce an STDP-like curve as a function of pre–post timing difference (Liu et al., 2019). BiSNN combines spiking dynamics with binary weights and introduces a Bayesian training rule over Bernoulli-distributed weights, improving uncertainty calibration relative to straight-through training while retaining competitive accuracy on MNIST-DVS (Jang et al., 2020).

6. Evaluation, limitations, and open problems

A persistent theme across BINN research is that biological structure must be evaluated mechanistically, not only predictively. In operator-learning settings, accurate state reconstruction $$\frac{\partial u}{\partial t} = \frac{\partial}{\partial x}\left(D(u)\frac{\partial u}{\partial x} - uV(u)\right) + uG(u),$$7 can coexist with poor recovered mechanisms, so forward simulation with the learned operators is the critical check (Crossley et al., 8 Jul 2026). In ABM coarse-graining, BINN-guided PDEs are valuable precisely because they remain predictive where analytic mean-field closures are ill-posed, but the neural training cost can exceed the cost of a single ABM run, making the approach most attractive when many evaluations across parameter space are needed (Nardini, 2023). In clinical segmentation, hierarchical conditioning improves interpretability and can yield competitive Dice scores, but downstream subnetworks inherit upstream WT errors and soft conditioning does not analytically guarantee containment or mutual exclusivity (Beers et al., 2017).

A common misconception is that adding biological prior structure is sufficient to guarantee superior accuracy. The available evidence does not support that claim. The glioma segmentation pipeline was competitive rather than dominant on BraTS (Beers et al., 2017). Mechanistic ADR studies show that reliable recovery requires balancing data and PDE losses, choosing moderate expressivity, and ensuring data informativeness over the relevant state range (Crossley et al., 8 Jul 2026). Genomic BINNs depend strongly on the quality of pathway annotations and on choosing an appropriate sparsity level; over- or under-sparsification can degrade performance, and transfer across genetically distant populations is limited when priors are too rigid (Kontolati et al., 16 Oct 2025). Process-guided soil-carbon BINNs inherit the structural assumptions of CLM5, including a steady-state approximation and potential equifinality among parameters (Xu et al., 2 Feb 2025).

The current research frontier is therefore not merely to add more biological detail, but to control the trade-off between inductive bias, identifiability, optimization stability, and uncertainty quantification. Several directions recur across the literature: explicit structured losses for subset and exclusivity constraints; likelihood-based or Bayesian treatments of noise and parameter uncertainty; extension from single-state systems to multi-species or higher-dimensional PDEs; hybrid symbolic-regression pipelines for closed-form law discovery; and larger architectures that combine pathway knowledge, connectomics, and brain-inspired learning rules (Crossley et al., 11 Jun 2026, Lavery et al., 20 Apr 2026). This suggests that the enduring value of BINNs lies less in any single implementation than in a modeling stance: biological knowledge is treated as a formal component of the hypothesis class, the training objective, or the optimizer, so that prediction and mechanism recovery are pursued jointly rather than sequentially.