Biomimetic PINNs for Cell-Induced Phase Transitions: UQ-R3 Sampling with Causal Gating

Abstract

Nonconvex multi-well energies in cell-induced phase transitions give rise to sharp interfaces, fine-scale microstructures, and distance-dependent inter-cell coupling, all of which pose significant challenges for physics-informed learning. Existing methods often suffer from over-smoothing in near-field patterns. To address this, we propose biomimetic physics-informed neural networks (Bio-PINNs), a variational framework that encodes temporal causality into explicit spatial causality via a progressive distance gate. Furthermore, Bio-PINNs leverage a deformation-uncertainty proxy for the interfacial length scale to target microstructure-prone regions, providing a computationally efficient alternative to explicit second-derivative regularization. We provide theoretical guarantees for the resulting uncertainty-driven ``retain-resample-release" adaptive collocation strategy, which ensures persistent coverage under gating and establishing a quantitative near-to-far growth bound. Across single- and multi-cell benchmarks, diverse separations, and various regularization regimes, Bio-PINNs consistently recover sharp transition layers and tether morphologies, significantly outperforming state-of-the-art adaptive and ungated baselines.

Figures (22)

• Figure 1: Overview of Bio-PINNs with distance-gated curriculum and UQ-R3 sampling. A logistic distance gate $g_\gamma(x)$ progressively activates the training domain from the near field to the far field, while a deformation-uncertainty proxy guides an R3 retain-resample-release update under a fixed collocation budget. The resulting closed-loop scheme concentrates samples on sharp layers and tether-forming regions across cell separations and regularization regimes.
• Figure 2: Mapping temporal causal gating to progressive distance gating. Left: a logistic time gate $g_{\beta}(t)=\sigma\!\left(\alpha(\beta-t)\right)$. Right: the corresponding distance gate $g_{\gamma}(\tilde{d})=\sigma\!\left(\alpha(\gamma-\tilde{d})\right)$, where $\tilde{d}$ denotes the normalized distance to the cell. Bottom: spatial snapshots of $g_{\gamma}(x)=\sigma\!\left(\alpha(\gamma-\tilde{d}(x))\right)$ for increasing $\gamma$, illustrating near-to-far activation from the cell neighborhood into the bulk.
• Figure 3: UQ-proxy evaluation via local probing. For each representative collocation point $x^\star$, Gaussian probes are sampled in a neighborhood of $x^\star$ and used to estimate local deformation variability (for example, $\operatorname{Var}_k\|\nabla y_{\theta_i^{(k)}}(x^\star)\|_F$). This yields a normalized score $\widetilde{U}_i(x^\star)\in[0,1]$ that highlights interface- and tether-forming regions.
• Figure 4: UQ-R3 retain-resample-release update. Step 1 (Resample): propose candidate collocation points within the current active region and, upon gate expansion, within the newly opened shell, using low-discrepancy designs. Step 2 (Evaluate $\rightarrow$ Retain/Release): evaluate the UQ proxy $\widetilde{U}_i$ and retain high-uncertainty points while releasing low-uncertainty points. Step 3 (Refill and iterate): resample to maintain a fixed collocation budget and repeat, progressively concentrating points near uncertain interfaces and emerging tethers while preserving global coverage.
• Figure 5: Single-cell benchmark without second-gradient regularization ($\varepsilon_0 = 0$). The Jacobian determinant $J=\det F$ (with $F=\nabla y_\theta$) is shown for (a) PINN, (b) RAR-D, (c) R3, and (d) Bio-PINN. Bio-PINN more consistently resolves the pericellular densified phase while reducing sampling-induced angular artifacts in the far-field ECM.

Theorems & Definitions (11)

• Proposition 5.1: Release: budget identity
• Theorem 5.2: Resample: non-emptiness and discrepancy-controlled coverage
• Theorem 5.3: Retain: hit, no-early-exit, and accumulation under good rounds
• Corollary 5.1: Guaranteed shell injection
• Theorem 5.4: Dynamic injection/accumulation under monotone gating
• Remark 5.1: Where proofs and stronger statements appear
• proof
• proof
• proof
• proof