Biologically informed neural network models are robust to spurious interactions via self-pruning

Biologically informed neural network models are robust to spurious interactions via self-pruning | bioRxiv /* */ /* */ <!-- <!-- /*! * yepnope1.5.4 * (c) WTFPL, GPLv2 */ (function(a,b,c){function d(a){return"[object Function]"==o.call(a)}function e(a){return"string"==typeof a}function f(){}function g(a){return!a||"loaded"==a||"complete"==a||"uninitialized"==a}function h(){var a=p.shift();q=1,a?a.t?m(function(){("c"==a.t?B.injectCss:B.injectJs)(a.s,0,a.a,a.x,a.e,1)},0):(a(),h()):q=0}function i(a,c,d,e,f,i,j){function k(b){if(!o&&g(l.readyState)&&(u.r=o=1,!q&&h(),l.onload=l.onreadystatechange=null,b)){"img"!=a&&m(function(){t.removeChild(l)},50);for(var d in y[c])y[c].hasOwnProperty(d)&&y[c][d].onload()}}var j=j||B.errorTimeout,l=b.createElement(a),o=0,r=0,u={t:d,s:c,e:f,a:i,x:j};1===y[c]&&(r=1,y[c]=[]),"object"==a?l.data=c:(l.src=c,l.type=a),l.width=l.height="0",l.onerror=l.onload=l.onreadystatechange=function(){k.call(this,r)},p.splice(e,0,u),"img"!=a&&(r||2===y[c]?(t.insertBefore(l,s?null:n),m(k,j)):y[c].push(l))}function j(a,b,c,d,f){return q=0,b=b||"j",e(a)?i("c"==b?v:u,a,b,this.i++,c,d,f):(p.splice(this.i++,0,a),1==p.length&&h()),this}function k(){var a=B;return a.loader={load:j,i:0},a}var l=b.documentElement,m=a.setTimeout,n=b.getElementsByTagName("script")[0],o={}.toString,p=[],q=0,r="MozAppearance"in l.style,s=r&&!!b.createRange().compareNode,t=s?l:n.parentNode,l=a.opera&&"[object Opera]"==o.call(a.opera),l=!!b.attachEvent&&!l,u=r?"object":l?"script":"img",v=l?"script":u,w=Array.isArray||function(a){return"[object Array]"==o.call(a)},x=[],y={},z={timeout:function(a,b){return b.length&&(a.timeout=b[0]),a}},A,B;B=function(a){function b(a){var a=a.split("!"),b=x.length,c=a.pop(),d=a.length,c={url:c,origUrl:c,prefixes:a},e,f,g;for(f=0;f<d;f++)g=a[f].split("="),(e=z[g.shift()])&&(c=e(c,g));for(f=0;f<b;f++)c=x[f](c);return c}function g(a,e,f,g,h){var i=b(a),j=i.autoCallback;i.url.split(".").pop().split("?").shift(),i.bypass||(e&&(e=d(e)?e:e[a]||e[g]||e[a.split("/").pop().split("?")[0]]),i.instead?i.instead(a,e,f,g,h):(y[i.url]?i.noexec=!0:y[i.url]=1,f.load(i.url,i.forceCSS||!i.forceJS&&"css"==i.url.split(".").pop().split("?").shift()?"c":c,i.noexec,i.attrs,i.timeout),(d(e)||d(j))&&f.load(function(){k(),e&&e(i.origUrl,h,g),j&&j(i.origUrl,h,g),y[i.url]=2})))}function h(a,b){function c(a,c){if(a){if(e(a))c||(j=function(){var a=[].slice.call(arguments);k.apply(this,a),l()}),g(a,j,b,0,h);else if(Object(a)===a)for(n in m=function(){var b=0,c;for(c in a)a.hasOwnProperty(c)&&b++;return b}(),a)a.hasOwnProperty(n)&&(!c&&!--m&&(d(j)?j=function(){var a=[].slice.call(arguments);k.apply(this,a),l()}:j[n]=function(a){return function(){var b=[].slice.call(arguments);a&&a.apply(this,b),l()}}(k[n])),g(a[n],j,b,n,h))}else!c&&l()}var h=!!a.test,i=a.load||a.both,j=a.callback||f,k=j,l=a.complete||f,m,n;c(h?a.yep:a.nope,!!i),i&&c(i)}var i,j,l=this.yepnope.loader;if(e(a))g(a,0,l,0);else if(w(a))for(i=0;i (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0];var j=d.createElement(s);var dl=l!='dataLayer'?'&l='+l:'';j.src='//www.googletagmanager.com/gtm.js?id='+i+dl;j.type='text/javascript';j.async=true;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-M677548'); Skip to main content Home About Submit ALERTS / RSS Search for this keyword Advanced Search New Results Biologically informed neural network models are robust to spurious interactions via self-pruning View ORCID Profile Olof Nordenstorm , View ORCID Profile Hratch Baghdassarian , View ORCID Profile Douglas A Lauffenburger , View ORCID Profile Avlant Nilsson doi: https://doi.org/10.1101/2025.10.24.684155 Olof Nordenstorm 1 Department of Cell and Molecular Biology, Karolinska Institutet , Stockholm, 17177, Sweden Find this author on Google Scholar Find this author on PubMed Search for this author on this site ORCID record for Olof Nordenstorm Hratch Baghdassarian 2 Department of Biological Engineering, Massachusetts Institute of Technology , Cambridge, MA 02139, USA Find this author on Google Scholar Find this author on PubMed Search for this author on this site ORCID record for Hratch Baghdassarian Douglas A Lauffenburger 2 Department of Biological Engineering, Massachusetts Institute of Technology , Cambridge, MA 02139, USA Find this author on Google Scholar Find this author on PubMed Search for this author on this site ORCID record for Douglas A Lauffenburger Avlant Nilsson 1 Department of Cell and Molecular Biology, Karolinska Institutet , Stockholm, 17177, Sweden Find this author on Google Scholar Find this author on PubMed Search for this author on this site ORCID record for Avlant Nilsson For correspondence: avlant.nilsson{at}ki.se Abstract Full Text Info/History Metrics Supplementary material Data/Code Preview PDF Abstract Computational models of cellular networks hold promise to uncover disease mechanisms and guide therapeutic strategies. Biology-informed neural networks (BINNs) is an emerging approach to create such models by combining the predictive power of deep learning with prior knowledge, a vital aspect of biological research. The architectures of BINN’s enforces a network structure from which mechanism can ideally be inferred. However, a key challenge is to evaluate the reliability of these mechanisms, as cells are inherently complex, involving intricate and sometimes unknown interactions. Currently, analysis mainly focuses on selected pathways rather than a more comprehensive perspective. In this work we demonstrate an improved, holistic approach: we measure to which extent purposefully introduced spurious interactions are removed by a BINN during training (self-pruning). This metric is scalable and generalizable, as it does not depend on manual curation and can be translated into diverse network settings. To enable rapid testing, we reimplemented LEMBAS (Large-scale knowledge-EMBedded Artificial Signaling-networks), our recurrent neural network framework for intracellular signaling dynamics, with full GPU acceleration. Our implementation achieves a >7-fold speedup compared to the original while preserving predictive accuracy. We evaluated self-pruning in 3 different datasets and found that when spurious interactions are introduced at random, the model prunes these to a larger extent than those from the prior knowledge network (PKN), provided the model is regularized with a sufficiently large L2 norm. This suggests that BINNs are robust to uncertainty in the PKN and is a quantitative sign that they could model real aspects of the systems. Our implementation of LEMBAS is freely available under a MIT license at https://github.com/AvlantNilssonLab/LEMBAS_GPU . The models and results to generate the figures can be downloaded through https://zenodo.org/records/17425598 . Introduction Intracellular signaling pathways allow cells to process and react to their environment. This includes transmission of signaling events from ligand-receptor interactions to transcription factors (TFs)( 1 ). Disruptions in signaling networks are common in diseases such as cancer( 2 – 4 ), and therefore, understanding the systemic effects of such perturbations is central to deciphering disease mechanisms and developing new drugs( 5 ). However, this is challenging for several reasons, including the vast combinatorial space of possible interaction in a signaling network, the non-linearity of these interactions, and inter-pathway signaling interactions across the network( 6 ). Therefore, computational modeling approaches have high utility for understanding signaling activity and the potential to transform therapeutic approaches by enabling rapid evaluation of interventions( 5 , 7 ). Traditionally, modeling approaches, such as ordinary differential equations (ODEs) and Boolean models, have been used to simulate cellular signaling( 8 ). These models are constructed using known molecular interactions in cells and thus serve not only as predictive tools but also have potential to reveal mechanistic relations. However, such models face limitations in scalability and expressiveness( 9 – 12 ). Specifically, ODE methods require detailed and comprehensive knowledge of the parameters describing molecular interaction; and when this is not available, these parameters are highly computationally expensive to infer( 13 ). Additionally, boolean models require extensive manual curation and are qualitative rather than quantitative by construction. As an alternative, artificial neural networks (ANNs) have emerged as a versatile and effective tool for approximating complex functions, with many biological applications( 14 ). Recently, ANNs that make use of prior biological knowledge have been developed. The aim of these models is to enable predictions that are (at least in principle) biologically interpretable and thereby bridge the gap between predictive accuracy and mechanistic insight. These models are termed Biologically Informed Neural Networks (BINNs)( 5 , 15 ), analogous to Physics-Informed Neural Networks (PINNs). Whereas PINNs embed physical knowledge (e.g., differential equations) into their loss functions or model design to ensure physical realism in outputs( 16 ), BINNs integrate biological priors such as interaction graphs into the architecture( 15 ). For example, we developed a BINN termed LEMBAS (Large-scale knowledge-EMBedded Artificial Signaling-networks) that uses a recurrent neural network to simulate intracellular signaling and explain steady-state TF activities in response to ligands and other perturbations( 17 ). It embeds known signaling interactions as structural constraints, mapping weights to real molecular interactions and setting all others to zero, making predictions interpretable and traceable( 17 , 19 ). Unlike conventional deep learning models with randomly initialized parameters with no correspondence to biological processes, LEMBAS links predictions to molecular mechanisms, offering a path towards constructing interpretable models of disease such as cancer ( 5 ). And in general, BINNs has been proven to offer mechanistic insight in many biological tasks( 17 – 20 ), suggesting that prior knowledge-informed networks could form a foundation for system biology analysis( 17 ). While BINNs hold promise to explain and predict biological data, several challenges that are associated with Graph Neural Networks (GNN) also translate to BINNs. One issue is misannotation of edges, which may be particularly challenging for biological systems where there are many methods to infer prior knowledge networks, and where the results can differ markedly, including spurious connections (false positives) as well as missing interactions (false negatives) ( 21 ). Failure to use the correct graph structure may cause incorrect relations to emerge in the network. We reason that spurious connections may be less detrimental, as GNN models can (at least in principle) down-weight (prune) interaction but most implementations cannot introduce new edges( 22 , 23 ), which also applies to BINNs( 17 , 18 ). However, the propensity of these networks to down weight incorrect edges through training on experimental data is typically not systematically evaluated. So far analysis of BINNs has mostly centered on its predictions for specific pathways ( 17 , 19 , 20 , 24 , 25 ), rather than on system level network properties. But analysis of the internal structure of deep learning networks can be a useful tool to understand how the algorithms behave during training( 26 , 27 ). An example of this is Grokking ( 28 ), a phenomenon where after training a model for many epochs following an overfit to the training data, there is rapid progress to a perfect generalization on the test set. This is hypothesized to be the result of the model finding an efficient algorithmic-like solution to fit the data( 29 , 30 ), and it has been observed that this is accompanied by a large fraction of the weights approaching zero, suggesting that the algorithmic solution constitutes a sparse network of strong interactions. We reason that this resembles the structured sparsity observed in biological networks and that therefore systems level analysis from the grokking literature may translate to the domain of BINNs. Here we propose a systems level metric for network soundness that tests if false edges introduced in the network are suppressed during training, thus mimicking false positives in prior knowledge. We argue that in practice, performing well on this test is a necessary, but insufficient, requirement for system-level trust in PKN-based models. To systematically analyze how a BINN performs on this metric, we implemented a GPU-accelerated version of LEMBAS, that preserves accuracy while reducing training time. We find that false edges were consistently assigned negligible weights and were thereby pruned over the course of training across synthetic and experimental datasets and across hyperparameter settings. While the work is inspired by grokking, where models shift from memorization to algorithmic solutions, we note that perfect recovery is unlikely in current BINN formulations, due to their extreme under parameterization. Additionally, there are structural differences between the formulations that may influence how training progresses. Indeed, we find that unlike the sharp transitions seen in grokking, pruning in this setting unfolded gradually. Overall, these results suggest BINNs are resilient to false interactions, supporting permissive prior inclusion and demonstrating their potential to infer mechanistic network structure. Results The reimplemented framework improves speed and preserves performance Extensive training and sampling is needed to test if a BINN framework is robust to false positives in the prior knowledge network. To enable this we reimplemented our BINN framework LEMBAS ( Figure 1a ) with GPUs capabilities, expecting a marked speed up. In conjunction, we also made modifications to align it with more conventional machine learning approaches to regularization terms in the loss function, as well as streamlining the overall structure of the code base (see Methods for details) . After the reimplementation we found a perfect correlation in outputs and gradients when only training with MSE (Supplementary Figure S1) , confirming that the core computations remained unchanged. However, we also updated the terms that enforced; steady state, state diversity, and our method to prevent confinement to local minima during training ( Figure 1b ). These changes are expected to cause discrepancies between the original CPU-based version of LEMBAS and the new, GPU-enabled version based on pytorch using CUDA( 31 ), and we therefore compared predictive performance along with computational speed of each implementation on two datasets that we previously used for training and benchmarking( 17 ). Both of these datasets measure the transcriptional response to ligand stimulation in macrophages, one with low coverage (23 unique stimulation conditions) ( 32 ) and one with high coverage (systematic stimulation using 59 different ligands, with and without co-stimulation with lipopolysaccharide) ( 17 ). Models trained on these datasets were tested using cross validation (leave one out, and leave three out), using the same folds as in the original publication. A dense matrix multiplication was used as opposed to the old sparse multiplication used in the signaling state. This is due to sparse matrix multiplications in GPU only benefit at higher level of sparsity that what we have in our networks that is >99% sparsity ( 33 ). Download figure Open in new tab Figure 1. Reimplemented BINN framework with improved speed and preserved performance. a) LEMBAS is designed to propagate a hidden state that mirrors the dynamics of signaling molecules following real signaling events. The input to the model is mapped into the hidden state and the output is extracted from the hidden states. b) The modifications to LEMBAS compatible with GPU include spectral, state, and network weight (i.e., node death prevention) regularization. The new architecture is flexible with regularization term changes. For example, the state prior and node death prevention regularizations can be used in both a GPU and CPU implementation; however, the method to calculate the spectral radius is specific to GPU or CPU, using power iteration and an eigen solver, respectively. c) Comparison of runtime between CPU and GPU implementation of LEMBAS on the low coverage dataset(N=23), using the Willcox test. The black line represents the mean for each of the data sets. d) a similar comparison for the high coverage data set using the Mann-Whitney U-test on the CPU (N=3) and GPU (N=28) model. e) Comparison of performance on leaving one in the low-coverage data set (N=23). f) Comparison of predictive performance using cross validation high-coverage data set with 28 folds. g) Comparison of the original hyper parameterization to combinations of new modules, a black box indicates the new version h) Comparing the different conditions to the reference model A. Each model weights is compared to a model having A conditions with the same random seed. For the low-coverage dataset, we preserved the hyperparameters from the CPU implementation. Here, runtime was reduced ( p < 0.001, Wilcoxon test) to 42% from ∼23 min to 9 min ( Figure 1c ), on a high end desktop computer (see hardware). However, we expected that a switch to the new implementation would affect the optimal hyperparameters for both predictive performance and speed of training, as regularization is modified and GPUs can handle large batches more efficiently, owing to the inherent capacity for parallelization. We therefore manually tuned the hyperparameters in the high-coverage dataset, with the parameter most drastically altered being the increased batch size. This resulted in 7.1x faster training compared to the original implementation ( p < 0.001, Mann-Whitney U-tes) from around 3 hours to 20 minutes ( Figure 1d ). Note that due to the long CPU runtime, we restricted the evaluation of runtime to three samples on the CPU and employed the Mann-Whitney U-test, which allows for unbalanced data sets. We reason that both analyses are relevant comparisons, but that the latter is more representative of the true speed gain, as using hyperparameters that were chosen to optimize performance on a distinct, old implementation with different regularization terms is not truly representative of how anyone would reasonably use a new implementation. We found that the predictive performance was consistent between the CPU and GPU version. In the low-coverage setting ( Figure 1f ) , the GPU implementation yielded a median Pearson correlation of 0.69 among the folds compared to 0.68 for the CPU implementation. Interestingly, this improvement was nearly significant (p-value of 0.056, Wilcoxon test), despite limited hyperparameter tuning. For the high-coverage dataset ( Figure 1e ) , the median performance was 0.83 for the GPU and 0.85 for the CPU implementation (n.s., Wilcoxon test p-value = 0.92), in line with our expectation that predictive performance will be maintained in the new GPU version. Notably, the GPU implementation improved the mean performance of the models: correlation increased from 0.55 to 0.64 (Cohen’s effect size 0.35) and from 0.79 to 0.82 (Cohen’s effect size 0.239) in the low- and high-coverage datasets respectively, potentially due to fewer outliers with poor performance. While the calculated effect size is small, we note that it is present in both datasets. The reimplemented framework preserves both output and gradients To follow up the observed differences in predictions, we investigated how the changes introduced affected output and gradients. One of these changes was that the CPU-dependent eigensolver for calculating the spectral radius loss, was replaced by a power iteration–based method previously used for RNNs( 34 – 37 ). We found that five iterations provided a near-perfect correlation (∼0.95) with the CPU-based solver (Supplementary Figure S2.) . We also introduced a more exact approximation of the for state regularization. This new implementation correlated perfectly (r=0.9996) with the original implementation but results in larger magnitudes (x1.483) given the same hyperparameters (Supplementary Figure S3) . We also observed in preliminary testing that steady state and state regularization were no longer necessary for stable training as they were in the original CPU implementation. While the exact reason for this remains unclear, it may potentially be due to changes incurred from the new eigenvalue calculations. This prompted us to explore how removing these would affect predictive performance and runtime on the low-coverage ligand dataset. While the model could be trained faster (without regularization 523 seconds with 577 seconds and a p value of p<0.0001) (Supplementary Figure S4) , predictive performance did not significantly decrease (p = 0.445, Willcox test). (Supplementary Figure S5) This suggests that inclusion of these regularizations can be considered optional in this implementation. Beyond assessing the direct effects of individual modifications, we also investigated their influence on learned parameters. We focused on three specific aspects: state regularization, weight regularization, and the inclusion of an additional bias on the output. The additional bias was, however, not used in the prior investigation, the bias has been introduced as an optional feature that can be included in the data and modelling demands it. These modifications represent true departures from the original implementation, in contrast to the power iteration method, which, if properly tuned, should reproduce the same eigenvalues as the original solver if so desired. No large departure in predictive accuracy was observed across all the conditions ( Figure 1g ) . While learned prior knowledge network weights remained largely unchanged following alterations to state regularization, adjustments to weight regularization induced a notable shift in the weight distributions. When both state and weight regularizations were modified simultaneously, deviations from the baseline were closest to the effect of weight regularization however it was slightly more pronounced. Notably, the effect of weight regularization appeared larger than that of introducing an additional output bias. While it could be expected that a modification to the model architecture would have a larger impact on parameters than a regularization term; the original L2 loss function was designed to strictly discourage zero weights and therefore its pronounced effects could be anticipated ( Figure 1h ) . A more detailed analysis comparing the weights of signaling (Supplementary Figure S6) and weights of the final output (Supplementary Figure S7) , shows that introduction of the output bias strongly affects the weights connecting the output of signaling to the model output, which is not surprising as the bias is then part of that output. False interactions vanish during training False positives in the context of biological graphs can be viewed as the existence of entries connecting biomolecules that do not directly interact. It is well known that modern methods to produce these interaction graphs can produce inconsistent and incorrect results, this manifest as incorrect signs (activation/inhibition) in interactions ( 38 ), and false positives and false negatives in the adjacency graph( 39 ). A neural network that is conditioned on such false interactions may potentially produce unrealistic structures that fail to model real biological processes. Ideally such false interactions should be removed during training, and it is standard practice in deep learning to simultaneously regularize weights (interactions) while optimizing the data fitting task. To investigate this self-pruning capacity of the framework, we added random edges to models and evaluate if these are pruned away during training ( Figure 2a ). For the sake of this analysis, we treat all stochastically added edges as false positives, even though theoretically the PKN may be missing interactions meaning that the added edges could be correct by accident in the real data setting. To test self-pruning in a setting where all added edges are certain to be false positives, we supplemented the 2 datasets previously described with an additional synthetic data set. It is generated using LEMBAS with the objective set to maximize signaling diversity within a network derived from KEGG( 40 ), and therefore has a known, and theoretically recoverable parameterization. Download figure Open in new tab Figure 2. Effects of weight pruning. a) Proposed approach for testing the effects of false positives. b) For each dataset, three different levels of L2 regularization were tested, and for each combination, 50 models were trained with 100 false interactions each. c) The effect of regularization on test performance in the different datasets. A white line indicates the best result for each dataset. d) Average deviation in output (black line) for increasing levels of weight pruning under the best performing setting. Performance of individual models traced in blue, the percentage of weights below threshold (dotted line) indicated. e) Average pruning level in each setting, quantified as the percent of added edges that were below the median. f) Weight pruning over the course of training in the high coverage dataset. The model state was saved at start, end, and 20 checkpoints through training. For each dataset we investigated the effects of 3 different levels of L2 regularization (low, medium, and high). To gain statistical confidence, we performed this analysis in an ensemble of 50 models, with 100 stochastically added edges each ( Figure 2b ). For all datasets the uniform regulation of state was removed, as its main purpose is to improve gradient flow of the model, not to enforce known priors about the signaling process, and therefore enforcing it could promote false correlation between nodes to satisfy this objective. It is known that a too low regularization level permits overfitting, while a too high-level leads to underfitting. We therefore evaluated the test performance under the different regularization settings and found that the optimal level is dataset dependent ( Figure 2c , Supplementary Figure 8 ). For the synthetic dataset, where most edges are expected to contribute to the solution, low regularization worked best, whilst high regularization worked best for the low coverage dataset, where only a subset of the network is expected to be activated. The high-coverage dataset achieved its optimal predictive performance in the medium regularization level; however, this performance was similar to the low level. To assess the self-pruning ability in these settings, we examined how low-magnitude weights contributed to the models ‘predictions. Our rationale for this was: if small weights meaningfully affect the output, then pruning based on weight norms would be misguided, as the magnitudes would not imply how meaningful an edge is. To test this, we performed an ablation study where all weights below a given threshold were set to zero. This allowed us to directly measure the influence of weak edges on predictive performance ( Figure 2d , Supplementary Figure S9) . We observed that a large fraction of weights can be removed with minimal performance loss, in many settings up to 50% of edges could be pruned. These findings support the idea that weight-based self-pruning is a meaningful metric in our framework. To track performance across training, we saved the model at regular intervals (checkpoints), along with the initial parameterization before training started. We recorded the weights of true and false edges and calculated their magnitude (absolute value). We then compared the percentage of the edges that were below the median. Overall, we found a strong overrepresentation of false interactions in the bottom 50% of weights ( Figure 2e ). The effects were generally larger for the experimental datasets than the synthetic data, and we reason that this may be due to the smaller network size in this setting where 100 edges compromise a much larger perturbation (12% vs 1.7%, and 1.5% for the synthetic, low- and high coverage dataset respectively). For the low-coverage data set self-pruning was strongly dependent on the regularization level, which stands in contrast to the test performance ( Figure 2d ) which was largely conserved, indicating that test performance and self-pruning, while conceptually similar, are to some degree decoupled. For the high-coverage dataset, we find that false interactions were not preferentially pruned in the low regularization setting. However, for the medium and high regularization settings the added weights are overrepresented among the weights below the median ( Figure 2f ) , of note is that optimal performance of the model is at the medium L2 level. For this analysis the median was used as a cutoff. However, since this is somewhat arbitrary, we also inspect the cumulative distributions of the true and false interactions for each data set and L2 level ( Figure 3 ). This was done in the middle point of the training trajectory (checkpoint 10), and the weights were fused across replicates to create a clear representation ( see methods ). We again find that the low L2 norm setting with high coverage data fails at the self pruning task, however for all other data sets and L2 settings we see that the cumulative density function (CDF) of added weights is generally larger than the CDF of the PKN weights. Download figure Open in new tab Figure 3. Cumulative density function (CDF) of weights for real and false interactions. For each dataset and regularization level the CDF was calculated for false and true edges independently at checkpoint 10, consistently showing a significant (Kolmogorov-Smirnov test) difference in the distributions. We highlight the places where the added weights have a larger CDF with a shaded red region. Discussion In this work, we provide evidence that biologically informed neural networks (BINNs) can exhibit an emergent self-pruning behavior. For this we implement a new GPU version of our BINN framework LEMBAS and find that it is faster to train and produces comparable results. Additionally, the GPU implementation appears to generate fewer outliers with poor performance resulting in better mean performance. This re-implementation aligned the framework better with more recent methods in machine learning, including support for CUDA and new design choices such as the gradient noise, paving the way for future work on LEMBAS with an additional speed up. Together, these improvements enable a systematic investigation of self-pruning as an emergent property of the training of a BINN, and we test this by stochastically adding false-positive edges to the PKN and tracking their behavior during training. For this we used a synthetic dataset with a known ground truth, and on two datasets from stimulated macrophages. Across all datasets, we found that the distribution of weights learned for false edges changed over the course of training, and that self-pruning was enhanced by increased levels of L2 regularization. Pruning and self-pruning is studied in the broader deep learning literature, particularly in the context of model compression( 41 ) and as a theoretical tool to understanding how deep learning models learn( 42 ). However, its emergence in the context of knowledge-informed network topologies has, to our knowledge, not been observed. This context introduces a key distinction: rather than pruning excess parameters in unconstrained architectures, here the network learns to denoise and refine biological prior knowledge, by actively down-weighting incorrect interactions in the knowledge network during training. When formulating this task, we were inspired by the results related to the grokking phenomenon. During Grokking, models trained on algorithmically simple tasks (e.g modular addition) ( 28 ) or on simple datasets (e.g classification on MNIST( 43 )) undergo a delayed phase transition: after an extended period of overfitting, the network abruptly discovers a generalizable, sparse representation that achieves near-perfect test accuracy. In these cases, it has been hypothesized ( 28 ) that the network synthesizes a compact algorithmic solution requiring only a small subset of weights, with the remainder becoming redundant and thus minimized. This hypothesis is supported by recent work ( 43 ) where the compact solution is particularly pronounced under L2 regularization, which also accelerates convergence by penalizing redundant parameters. In this work we observed that the training dynamics follow a more typical pattern: generalization improves steadily, and self-pruning appears to emerge as a by-product of regular training, potentially due to the sparsity already being enforced at the beginning of training. Nevertheless, it indicates that the redundant parameters introduced in addition to the true biological structure are being removed throughout training. A key limitation in comparing BINNs to the systems normally studied in the Grokking literature stems from the intrinsic complexity of cells. Unlike synthetic tasks with compact solutions, cellular signaling is not expected to conform to the simplified functional forms of current BINNs such as LEMBAS. This relates to two main limitations: only a subset of all cellular processes are included in the models; and the biological mechanisms themselves are far more intricate than their representations. For example, in LEMBAS the contribution of each interaction is modeled independently, ignoring the interdependent relations between molecules. Another example is subcellular compartmentalization, which is ignored in LEMBAS but critical in real cells. As a result, the model is heavily under parameterized. We reason that the limited self-pruning observed in high-coverage, low-L2 datasets may reflect a scenario where the extra parameters introduced partially compensate for this lack of complexity. Conversely, when training on synthetic data, the model shows stronger pruning, consistent with its well-parameterized functional structure. It is therefore encouraging that self-pruning emerges despite these constraints, suggesting that BINNs still learn meaningful approximations to underlying biological mechanisms. This points to their potential not only as predictive tools but also as engines for hypothesis testing and rejection in cellular systems. Here, the architectural structure is more important than the exact parameter count: a large, dense network without biological grounding, while technically not under parameterized, may fit data but would remain opaque and less mechanistically informative. So, to meaningfully mirror biological systems and learn mechanistically plausible representations, BINNs must strike a balance between expressiveness and structural constraints. As highlighted in many recent articles ( 5 , 7 , 15 ), the field of BINNs is in need of robust evaluation frameworks and systematic strategies for hypothesis generation. We propose that the emergence of self-pruning holds promise of progress for both challenges. Specifically, the selective removal of non-contributory or biologically implausible edges during training can serve as an implicit validation mechanism. A framework that fails to prune stochastic edges and instead relies on them for prediction, may indicate a failure to replicate biological reality, that would likely extend to false-positive interactions in the prior knowledge network (PKN). Frameworks that promote self-pruning enable a more permissive initial construction of PKNs that include uncertain interactions, with the expectation that training will refine this structure. Importantly, systems level metrics such as the one proposed here provide a simple, architecture-agnostic test that does not rely on manual pathway curation, unlike many existing tests of biological feasibility. However, we emphasize that self-pruning alone is limited in scope and should be combined with complementary metrics to assess how well models approximate the real mechanism inside of cells. Methods Datasets Throughout this paper we make use of three data sets: a high-coverage( 17 ), low-coverage( 32 ) and synthetic data set( 17 ). The high and low coverage data set are both instances of macrophages simulated with ligands with a read out of gene expression. The synthetic data set was created using the CPU implementation of LEMBAS. Matrix multiplications and gradient calculations We replaced the sparse matrix computations that were used in the original implementation of LEMBAS with dense matrix computations performed on a GPU, using python (v3.12.3) pytorch (v2.4.0), and cuda (v12.4). Originally, LEMBAS used accelerated sparse matrix computations that are possible on CPU, but the porting to a GPU required the use of dense operations. To ensure that only allowed interactions are maintained throughout training with the dense implementation, we expand our weight-update rule to set non-existing interaction to zero after each weights update has been performed. To make use of the sparse structure of the prior knowledge network, the original implementation of LEMBAS used a manually derived implementation of backpropagation and gradient calculation. The GPU-enabled version instead makes use of the standard autograd method that is integrated in pytorch. Spectral radius computation Originally, a spectral radius loss term was introduced to ensure convergence to a steady state. For this, sparse eigenvalue calculations were performed using the scipy library, which is based on the ARnoldi PACKage (ARPACK) implementation( 45 ). This method was not available on the GPU, and we replaced it with a power iteration method to approximate the dominant eigenvalue directly on the GPU. Instead of checking for convergence at each step of the algorithm, a fixed number of iterations ( 5 ) was used, as this reduces the computational overhead. Weight norm regularization and gradient noise In this implementation, we penalize the model weights using standard L2 regularization. The original implementation included an additional term that also penalized weights near zero to prevent neuron death. In this implementation, neuron death is instead addressed by injection of gradient noise. We set the default regularization strength of this term to be the same as in the original CPU implementation (λ= 1e-6). To mitigate neuron death during training and potentially improve generalization, we applied gradient noise injection to parameters during training( 46 ). The noise was drawn from a standard normal distribution scaled by a tunable hyperparameter. Formally, if g is the original gradient and ϵ∼N(0,I), the updated gradient becomes g′=g+γϵ where γ is the scaling hyperparameter. We set γ = 1e-9 for all analyses, except for cross validation on the hyperparameter optimized high-coverage ligand dataset, in which a varying noise scheme was used. Specifically, the noise was scaled according to the current learning rate: g′=g+αγϵ where α is the learning rate for this setup and we set γ = 1e-6. To measure the similarity of effect of the new regularization implementation on the steady state, a correlation was calculated on the gradient between the two implementations periodically (every 5 epochs) throughout training. State regularization LEMBAS regularizes the predicted output to penalize deviation from a uniform distribution. This state regularization serves as a bayesian prior of diversity between samples. The original implementation penalized the mean and variance of a uniform distribution.The new implementation uses evenly spaced values (via linspace) to approximate a uniform distribution prior more directly. That is, we approximate the result of uniform distribution and penalize the deviation of the output from this distribution using MSE. We also penalize output values that fall outside the unit interval in alignment with the original implementation. A regularization strength of λ= 1e-6 was used for all analyses. To evaluate the similarity in the state regularization the gradients were stored and a final correlation was calculated at the end of the training. Addition of bias term on output To enhance model flexibility and address potential data biases, an optional bias term was incorporated in the output layer of LEMBAS. This removes the non-negativity requirement on the model output, bypassing the need to apply additional transformation on inferred or measured TF activities, which may include negative values. As such, the predicted TF activity can be matched up directly with the data. Other minor modifications The gradient clipping term was changed to a norm gradient as it is a less biased scaling step.The frequency of convergence checks within the signaling network’s RNN propagation was reduced from being performed every iteration to every 10 iterations. This is due to the check requiring an expensive synchronization operation in the GPU setting. Performance Evaluation and Comparisons We employed Cross-Validation (CV) to evaluate the effect of each algorithmic modification, independently, and together. Performance was assessed using Pearson correlation between predicted values and data. Execution time was assessed by the wall time for each fold in the CV. For the low-coverage dataset, the original hyperparameters were used to isolate for effects caused by the GPU implementation directly. For the high-coverage dataset, hyperparameter optimization was manually performed to decrease computational time while retaining predictive performance. This focused mainly on gradient noise schedules and batch size as these were found to drive most of the variance in performance. For the high-coverage dataset the CPU version of LEMBAS was run 3 times and used for the time comparison, for CV the reported correlations from the original publication were used. Self-pruning To assess the sensitivity of the signaling module to weight pruning. We investigate how big of an effect removing all weights lower than a specific cutoff has on the final output. We trained an ensemble of models for each dataset and level of L2 regularization, adding 100 stochastic edges to each PKN. The model weights were saved at checkpoints throughout training. The CDFs of edge weights were retrieved at the middle checkpoint. The hyperparameters for the model were the same as for the cross validation study. For the synthetic data set we reused the low-coverage models pick of hyperparameters, with state regularization removed. Hardware for simulations Walltime comparison and CV on the low-coverage and high-coverage ligand dataset were performed on a Dell Precision 3680 with Intel(R) core(™) i7-14700 cpu and an RTX 4000 Ada Generation Graphics Card. For the generation of self-pruning and cross validation on the high-coverage ligand dataset we used the Berzelius AI/ML cluster using several different GPUs. We also used Satori a computational resource supplied by MIT. Satori is a GPU dense, high-performance Power 9 system developed as a collaboration between MIT and IBM. It has 64 1TB memory Power 9 nodes. Each node hosts four NVidia V100 32GB memory GPU cards. Within a node GPUs are linked by an NVLink2 network that supports nearly 200GB/s bi-directional transfer between GPUs. A 100Gb/s Infiniband network with microsecond user space latency connects the cluster nodes together. Availability and Implementation Our implementation of LEMBAS is freely available under a MIT license at https://github.com/AvlantNilssonLab/LEMBAS_GPU . The models and results to generate the figures can be downloaded through https://zenodo.org/records/17425598 . Acknowledgements We acknowledge funding from the SciLifeLab & Wallenberg Data-Driven Life Science Program grant no. KAW 2020.0239 (O.N, A.N.). The work of H.M.B and D.A.L. was supported in part by NIH grants (IMPAcTB contract #75N93019C00071, U19-AI167899, U19-AI135995). H.M.B. was supported by the MIT-Novo Nordisk Artificial Intelligence Postdoctoral Fellows Program and a Cancer Research Institute Immuno-Informatics Postdoctoral Fellowship (CRI12812). The computations were enabled by the supercomputing resource Berzelius provided by the National Supercomputer Centre at Linköping University and the Knut and Alice Wallenberg foundation. Funder Information Declared Knut and Alice Wallenberg Foundation, https://ror.org/004hzzk67 , 2020.0239 NIH Common Fund , IMPAcTB #75N93019C00071 , U19-AI167899 , U19-AI135995 MIT-Novo Nordisk Artificial Intelligence Postdoctoral Fellows Program , CRI12812 Footnotes Update to remove comment on sources that was attached on the orginal pdf. https://github.com/AvlantNilssonLab/LEMBAS_GPU https://zenodo.org/records/17425598 References 1. ↵ Kholodenko BN . Cell-signalling dynamics in time and space . Nat Rev Mol Cell Biol . 2006 Mar ; 7 ( 3 ): 165 – 76 . OpenUrl CrossRef PubMed Web of Science 2. ↵ Sever R , Brugge JS . Signal Transduction in Cancer . Cold Spring Harb Perspect Med . 2015 Apr 1; 5 ( 4 ): a006098 – a006098 . OpenUrl Abstract / FREE Full Text 3. Zhong L , Li Y , Xiong L , Wang W , Wu M , Yuan T , et al. Small molecules in targeted cancer therapy: advances, challenges, and future perspectives . Signal Transduct Target Ther . 2021 May 31; 6 ( 1 ): 201 . 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Share Biologically informed neural network models are robust to spurious interactions via self-pruning Olof Nordenstorm , Hratch Baghdassarian , Douglas A Lauffenburger , Avlant Nilsson bioRxiv 2025.10.24.684155; doi: https://doi.org/10.1101/2025.10.24.684155 Share This Article: Copy Citation Tools Biologically informed neural network models are robust to spurious interactions via self-pruning Olof Nordenstorm , Hratch Baghdassarian , Douglas A Lauffenburger , Avlant Nilsson bioRxiv 2025.10.24.684155; doi: https://doi.org/10.1101/2025.10.24.684155 Citation Manager Formats BibTeX Bookends EasyBib EndNote (tagged) EndNote 8 (xml) Medlars Mendeley Papers RefWorks Tagged Ref Manager RIS Zotero Tweet Widget Facebook Like Google Plus One Subject Area Systems Biology Subject Areas All Articles Animal Behavior and Cognition (7637) Biochemistry (17705) Bioengineering (13899) Bioinformatics (41968) Biophysics (21460) Cancer Biology (18603) Cell Biology (25526) Clinical Trials (138) Developmental Biology (13385) Ecology (19909) Epidemiology (2067) Evolutionary Biology (24326) Genetics (15614) Genomics (22513) Immunology (17741) Microbiology (40423) Molecular Biology (17193) Neuroscience (88645) Paleontology (667) Pathology (2835) Pharmacology and Toxicology (4825) Physiology (7647) Plant Biology (15160) Scientific Communication and Education (2046) Synthetic Biology (4302) Systems Biology (9825) Zoology (2271)