Knowledge-Inclusive Adaptive Physics-Informed Neural Network for Microbial Interaction Modelling

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 $1\mathrm{D}{+}t$ 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 $2\mathrm{D}{+}t$ 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 $2\mathrm{D}{+}t$ 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.

Amortizing physics-informed neural networks (PINNs) across related PDEs requires describing each equation to a reusable solver. Coefficient vectors encode numerical parameters in predefined slots, leaving operator and cross-field assignments implicit. We make these relationships explicit in an operator graph, with nodes for fields, derivatives, terms, and residuals and coefficients retained as term attributes. A graph hypernetwork generates diagonal codes that initialize a meta-trained factorized PINN for each target equation. Meta-training and target-specific adaptation use governing equations and prescribed conditions without solution labels. We compare coefficient-vector, DeepSets-based term-set, and graph conditioning by solution accuracy within a fixed adaptation budget. In scalar convection-diffusion-reaction problems, both term-based descriptors improve high-reaction accuracy, with similar performance. In two-field Fisher-KPP, meta-training sees uncoupled and one-way systems; after 3,000 adaptation steps on unseen two-way coupling, the graph's mean final error is 35.7% below the term set and 67.7% below the coefficient vector. In a fixed-structure capacitively coupled plasma model, the coefficient vector performs best. These results support extending coefficient conditioning with explicit equation relationships for physics-based solver adaptation.

Solving inverse problems in dynamical systems governed by high-dimensional coupled ordinary differential equations (ODEs) is a ubiquitous challenge in scientific machine learning. In many real-world applications, researchers seek to uncover unknown parameters or model unknown dynamics even as the underlying physics is only partially characterized, and observations are sparse and limited to specific measurable channels. While physics-informed neural networks (PINNs) are ideal for inverse inference under partial observability, existing PINNs typically rely on task-specific joint optimization, which suffers from optimization difficulties and poor generalization. In this paper, we propose a meta-inverse physics-informed neural network (MI-PINN) that reformulates inverse modeling as a two-stage meta-learning problem. MI-PINN first learns a physics-aware representation across multiple tasks, and then performs inverse modeling by optimizing task-specific unknowns while keeping the learned representation fixed. This two-stage formulation significantly reduces the parameter search dimension, thereby improving sample efficiency and enabling accurate inference. To handle multi-scale dynamics common in these high-dimensional ODE systems, we further introduce an adaptive clustering-based multi-branch learning scheme. We demonstrate the effectiveness of MI-PINN on whole-body physiologically based pharmacokinetic (PBPK) models with up to 33 coupled ODEs, using paracetamol and theophylline under intravenous and oral dosing scenarios. Experimental results show that MI-PINN enables accurate recovery of masked kinetic parameters and reconstruction of missing mechanistic terms despite limited clinical observations.