Cell-induced densification and tether formation in fibrous extracellular matrices with biomimetic physics-informed neural networks
Authors: Anci Lin, Zhiwen Zhang, Wenju Zhao
Organizations: School of Mathematics, Shandong University, Jinan, Shandong, 250100, P.R. China. · Department of Mathematics, The University of Hong Kong, Pokfulam Road, Hong Kong, Hong Kong SAR, P.R. China. · Materials Innovation Institute for Life Sciences and Energy (MILES), HKU-SIRI, Shenzhen, P.R. China.
Nonconvex multi-well energies in cell-induced phase transitions give rise to fine-scale microstructures, low-regularity transition layers and sharp interfaces, all of which pose numerical challenges for physics-informed learning. Here we introduce biomimetic physics-informed neural networks (Bio-PINNs), which implement a near-to-far curriculum by progressively revealing the computational domain away from the cell boundary and combining this schedule with a deformation-uncertainty proxy that concentrates collocation points near evolving transition layers and tether-forming regions. Across single-cell and multicellular benchmarks, Bio-PINNs recover the densified phase more reliably near cell boundaries and in intercellular gaps, while capturing tether morphology more faithfully than representative ungated and residual-driven adaptive baselines.
Figures & tables
Figure 1: Experiment–simulation comparison for mechanically interacting acini. (a) Fluorescence image of two mammary acini embedded in collagen, adapted from Fig. 2B of Shi et al. Shi et al. (2014) (included here with permission from the authors). (b) Bio-PINN solution in the long-distance weak-coupling regime, d=5rc . The numerical panel mirrors the visual structure of the experimental panel: Zoom 1 is placed inside the main field at the bottom and highlights the intercellular corridor, whereas Zoom 2 is placed externally on the right and highlights the pericellular band on the outer cell flank. Both panels display the coexistence of an intercellular mechanically organized corridor and an outer pericellular band. In the numerical panel, colour denotes J=detF , with smaller J indicating stronger densification.
Figure 2: Cell-induced remodelling motivates the Bio-PINN workflow. A logistic distance gate gγ(x) progressively exposes the matrix from the near-cell region to the far field, while a deformation-uncertainty proxy drives an R3 retain, resample, and release update under a fixed collocation budget. Together, these steps focus training on pericellular transition layers and tether-forming regions as the active domain expands.
Figure 3: Near-to-far activation derived from the physical progression of remodelling. The left panel shows a logistic time gate gβ(t)=σ(α(β−t)) . The right panel shows its distance-based analogue gγ(d~)=σ(α(γ−d~)) , where d~ is the normalized distance to the cell. The bottom panels show spatial snapshots of gγ(x)=σ(α(γ−d~(x))) for increasing γ , showing how the active region expands from the pericellular neighbourhood into the surrounding matrix.
Figure 4: Local probing used to detect regions likely to develop transition layers or tethers. Around each representative collocation point x⋆ , Gaussian probes sample nearby deformations and estimate local variability, for example Vark∥∇yθi(k)(x⋆)∥F . The resulting normalized score Ui(x⋆)∈[0,1] is largest near evolving interfaces and intercellular interaction regions.
Figure 5: Adaptive collocation loop used in Bio-PINNs. In Step 1, candidate collocation points are proposed in the currently active region and, after gate expansion, in the newly opened shell using low-discrepancy designs. In Step 2, the UQ proxy Ui is evaluated and high-uncertainty points are retained while low-uncertainty points are released. In Step 3, the collocation set is refilled to preserve a fixed budget and the procedure is repeated, so that sampling progressively concentrates on transition layers and emerging tethers while maintaining coverage of the active domain.
Figure 6: Single-cell remodelling without second-gradient regularization ( ε0=0 ). The Jacobian determinant J=detF (with F=∇yθ ) is shown for (a) PINN, (b) RAR-D, (c) R3, and (d) Bio-PINN. The benchmark asks whether the method can recover a densified pericellular band while leaving the surrounding matrix close to the undensified state. Among the methods shown here, Bio-PINN yields the lowest near-cell J values together with a more azimuthally uniform pericellular band.
Figure 7: Single-cell remodelling with weak second-gradient regularization ( ε0=0.01rc ). The Jacobian determinant J=detF (with F=∇yθ ) is shown for (a) PINN, (b) RAR-D, (c) residual-driven R3, and (d) Bio-PINN. The weak H2 penalty smooths the finest oscillations across methods, but Bio-PINN still yields a more coherent pericellular band and lower near-cell J values than the baselines shown here.
Figure 8: Two-cell remodelling without second-gradient regularization ( ε0=0 ). The Jacobian determinant J=detF (with F=∇yθ ) is shown for a long-distance regime (top row; d=5rc , weak coupling) and a short-distance regime (bottom row; d=2.5rc , strong coupling). The long-distance case probes weak coupling between cells, whereas the short-distance case probes tether formation in the intercellular gap. Panels (a,e) correspond to PINN, (b,f) to RAR-D, (c,g) to residual-driven R3, and (d,h) to Bio-PINN. Among the methods shown here, Bio-PINN produces a more contiguous densified band next to each cell and in the interaction region, with less spurious structure in the surrounding matrix. In particular, panel (d) recovers the same pair of motifs emphasized in Fig. 1 : an intercellular corridor and an outer pericellular band, whereas panel (h) sharpens the intercellular low- J region into a tether.
Figure 9: Three-cell remodelling stress test without second-gradient regularization ( ε0=0 ). The Jacobian determinant J=detF (with F=∇yθ ) is compared across methods for a long-distance regime (top row; weak coupling) and a short-distance regime (bottom row; strong coupling). The colour scale indicates whether the densified phase is reached around each cell and in the multicell interaction core. Bio-PINN reaches lower near-cell and interaction-core J values than the baselines while keeping the surrounding matrix closer to the undensified state.
Appendix figures & tables15 assets
Supplementary material from the paper’s appendix.
Appendix
Component
Default setting
Network
3 hidden layers, width 128, CELU activation
Optimizer
Adam ( 10−3 , β1=0.9 , β2=0.999 ); LR ×0.9 /10k iters
Table S1 : Default hyperparameters used in all experiments. Network architecture, optimizer settings, collocation budget, quadrature, gating schedule, R3 policy, and UQ-proxy parameters.
Figure S1 : Effect of the resampling period P in the single-cell non-regularized regime ( ε0=0 ). Each panel shows the Jacobian determinant J=detF (with F=∇yθ ).
Figure S2 : Effect of network depth hl in the single-cell non-regularized regime ( ε0=0 ). Each panel shows the Jacobian determinant J=detF (with F=∇yθ ) for different hidden-layer depths.
Figure S3 : Effect of network width in the single-cell non-regularized regime ( ε0=0 ). Each panel shows the Jacobian determinant J=detF (with F=∇yθ ) for different hidden-layer widths.
Figure S4 : Effect of the UQ probe count muq in the single-cell non-regularized regime ( ε0=0 ). Each panel shows the Jacobian determinant J=detF (with F=∇yθ ).
Figure S5 : Effect of the UQ probe variance ρuq in the single-cell non-regularized regime ( ε0=0 ). Each panel shows the Jacobian determinant J=detF (with F=∇yθ ).
Figure S6 : Effect of the resampling period P in the single-cell weakly regularized regime ( ε0=0.01rc ). Each panel shows the Jacobian determinant J=detF (with F=∇yθ ).
Figure S7 : Effect of network depth hl in the single-cell weakly regularized regime ( ε0=0.01rc ). Each panel shows the Jacobian determinant J=detF (with F=∇yθ ) for different hidden-layer depths.
Figure S8 : Effect of network width in the single-cell weakly regularized regime ( ε0=0.01rc ). Each panel shows the Jacobian determinant J=detF (with F=∇yθ ) for different hidden-layer widths.
Figure S9 : Effect of the resampling period P in the two-cell non-regularized regime ( ε0=0 ). The Jacobian determinant J=detF (with F=∇yθ ) is shown for the short-distance regime ( d=2.5rc ; top) and the long-distance regime ( d=5rc ; bottom).
Figure S10 : Effect of network depth hl in the two-cell non-regularized regime ( ε0=0 ). The Jacobian determinant J=detF (with F=∇yθ ) is shown for the short-distance regime ( d=2.5rc ; top row) and the long-distance regime ( d=5rc ; bottom row).
Figure S11 : Effect of network width in the two-cell non-regularized regime ( ε0=0 ). The Jacobian determinant J=detF (with F=∇yθ ) is shown for the short-distance regime ( d=2.5rc ; top row) and the long-distance regime ( d=5rc ; bottom row).
Figure S12 : Effect of the UQ probe variance ρuq in the two-cell non-regularized regime ( ε0=0 ). The Jacobian determinant J=detF (with F=∇yθ ) is shown for the short-distance regime ( d=2.5rc ; top row) and the long-distance regime ( d=5rc ; bottom row).
Figure S13 : Effect of the resampling period P in the three-cell non-regularized regime ( ε0=0 ). The Jacobian determinant J=detF (with F=∇yθ ) is shown for the short-distance regime ( d=2.5rc ; top row) and the long-distance regime ( d=5rc ; bottom row).
Figure S14 : Effect of the UQ probe variance ρuq in the three-cell non-regularized regime ( ε0=0 ). The Jacobian determinant J=detF (with F=∇yθ ) is shown for the short-distance regime ( d=2.5rc ; top row) and the long-distance regime ( d=5rc ; bottom row).
Physics-informed neural networks (PINNs) provide a powerful framework for learning governing equations of dynamical systems from data. Biologically-informed neural networks (BINNs) are a variant of PINNs that preserve the known differential operator structure (e.g., reaction-diffusion) while learning constitutive terms via trainable neural subnetworks, enforced through soft residual penalties. Existing BINN studies are limited to 1D+t reaction-diffusion systems and focus on forward prediction, using the governing partial differential equation as a regulariser rather than an explicit identification target. Here, we extend BINNs to 2D+t systems within a PINN framework that combines data preprocessing, BINN-based equation learning, and symbolic regression post-processing for closed-form equation discovery. We demonstrate the framework's real-world applicability by learning the governing equations of lung cancer cell population dynamics from time-lapse microscopy data, recovering 2D+t reaction-diffusion models from experimental observations. The proposed framework is readily applicable to other spatio-temporal systems, providing a practical and interpretable tool for fast analytic equation discovery from data.
William Lavery, Jodie A. Cochrane, Christian Olesen +3
Dept. of Information Technology Uppsala University Uppsala, Sweden · Dept. of Medical Genetics Oslo University Hospital Oslo, Norway · Dept. of Translational Hematology and Oncology Research, Cleveland Clinic, Cleveland, USA +1
Physics-informed neural networks (PINNs) are a versatile tool in the burgeoning field of scientific machine learning for solving partial differential equations (PDEs). However, determining suitable training strategies for them is not obvious, with the result that they typically yield moderately accurate solutions. In this article, we propose a method for improving the accuracy of PINNs by coupling them with a post-processing step that seeks the best approximation in a function space associated with the network. We find that our method yields errors four to five orders of magnitude lower than those of the parent PINNs across architectures and dimensions. Moreover, we can reuse the basis functions for the linear space in more complex settings, such as time-dependent and nonlinear problems, allowing for transfer learning. Our approach also provides a residual-based metric that allows us to optimally choose the number of basis functions employed.
Saad Qadeer, Panos Stinis
Advanced Computing, Mathematics and Data Division, Pacific Northwest National Laboratory, Richland, WA
In solving partial differential equations (PDEs), machine learning utilizing physical laws has received considerable attention owing to advantages such as mesh-free solutions, unsupervised learning, and feasibility for solving high-dimensional problems. An effective approach is based on physics-informed neural networks (PINNs), which are based on deep neural networks known for their excellent performance in various academic and industrial applications. However, PINNs struggled with model training owing to significantly slow convergence because of a spectral bias problem. In this study, we propose a PINN-based method equipped with a coordinate-encoding layer on linear grid cells. The proposed method improves the training convergence speed by separating the local domains using grid cells. Moreover, it reduces the overall computational cost by using axis-independent linear grid cells. The method also achieves efficient and stable model training by adequately interpolating the encoded coordinates between grid points using natural cubic splines, which guarantees continuous derivative functions of the model computed for the loss functions. The results of numerical experiments demonstrate the effective performance and efficient training convergence speed of the proposed method.
Tetsuro Tsuchino, Motoki Shiga
Graduate School of Engineering, Gifu University, Japan · International Center for Synchrotron Radiation Innovation Smart, Tohoku University, Japan · Unprecedented-scale Data Analytics Center, Tohoku University, Japan +3