Meta-Inverse Physics-Informed Neural Networks for High-Dimensional Ordinary Differential Equations

Abstract: 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.

• The paper presents a meta-learning framework that decouples reusable representation learning from task-specific inverse optimization to address challenges in high-dimensional ODE systems.
• It utilizes a multi-branch clustering strategy and a physics-informed pseudo-inverse readout to substantially reduce mean squared error and overcome scalability issues of conventional PINNs.
• Experimental results on PBPK models demonstrate accurate parameter inference and mechanistic discovery with notable computational speed-up and robust performance under sparse data.

Meta-Inverse Physics-Informed Neural Networks for High-Dimensional ODEs

Introduction

The paper "Meta-Inverse Physics-Informed Neural Networks for High-Dimensional Ordinary Differential Equations" (2605.03511) systematically addresses the challenges posed by inverse problems in high-dimensional coupled ODE systems, where both strong coupling and severe partial observability are present. Classical PINN-based approaches to inverse modeling exhibit poor scalability when deployed in large dynamical systems subject to multi-scale phenomena and limited measured outputs, due to entangled optimization of both network and task-specific parameters. The authors propose a meta-inverse physics-informed neural network (MI-PINN) methodology that formalizes parameter inference and mechanistic discovery under ODE constraints as a meta-learning problem, thereby decoupling reusable representation learning from task-specific inverse optimization.

Methodological Framework

Meta-Inverse Learning Formulation

MI-PINN introduces a two-stage learning paradigm. In Stage I, meta-representation learning is performed across a distribution of tasks, each defined by varying parameter or mechanistic configurations. A shared, physics-aware neural network representation is optimized, leveraging an adaptive clustering-based multi-branch architecture to address spectral bias and capture compartment-specific multi-scale features. Importantly, the final readout layers are updated not via standard backpropagation, but through a physics-informed closed-form pseudo-inverse solution, which stabilizes the meta-learning process by regularizing the reconstruction in the least-squares sense.

In Stage II, the learned meta-representation is frozen and only the task-specific unknown parameters are optimized for the target system, using available observations. The decoupling reduces the search space dimensionality from $N_e + N_a$ (network and parameter dimensions) to $N_a$, significantly improving identifiability and sample efficiency, especially when $N_e \gg N_a$. The closed-form final layer update is maintained during this stage, supporting stable optimization even under noisy or sparse data constraints.

Multi-Branch Meta-Representation Learning

To address the intrinsic heterogeneity of high-dimensional ODE systems, the authors cluster state variables based on dynamical similarity and assign dedicated network branches to each cluster. This specialization mitigates interference between disparate dynamic regimes and directly counters pathologies arising from spectral bias in PINN training. The approach systematically outperforms single-branch architectures, as demonstrated by an order-of-magnitude reduction in mean-squared error (MSE) in ablation studies.

Physics-Informed Pseudo-Inverse Readout

By representing the output layer weights as solutions to a physics-informed regularized least-squares problem (incorporating both data-fitting and ODE residuals), the final layer admits an analytical solution at each iteration. For nonlinear dependencies, the lagged-coefficient scheme is applied to linearize locally, enabling iterative refinement of parameter estimates. This removed the instability of gradient-based updates at the final layer and yielded improved conditioning in the inverse problem.

Experimental Validation

Large-Scale PBPK Application

The efficacy of MI-PINN is validated on whole-body physiologically based pharmacokinetic (PBPK) models of paracetamol and theophylline, involving systems up to 33 coupled ODEs. These models serve as archetypal benchmarks due to severe partial observability (often only the venous concentration is measured) and strong compartmental coupling.

Parameter Inference:

• In both IV and oral paracetamol scenarios, MI-PINN accurately infers masked kinetic parameters (e.g., $V_{max}$, $K_m$ for UGT2B15 metabolism) from as few as 10 observation points.
• Comparative results show parameter estimates within $\sim20\%$ of ground truth, outperforming conventional solver-based optimization and baseline PINNs, especially in the high-dimensional settings where traditional algorithms yielded erratic or inflated parameter estimates despite similar catalytic efficiency ratios.

Missing Mechanism Discovery:

• The authors demonstrate effective recovery of a masked metabolic term by training an auxiliary network within MI-PINN, followed by symbolic regression and dimensional analysis. The reconstructed function closely approximated the correct Michaelis-Menten structure, despite limited identifiability of negligible nonlinear terms in practice—highlighting the framework's potential for data-driven scientific discovery.

Efficiency Benchmarks:

• A single MI-PINN forward inference achieved a $\sim3.5\times$ speedup over direct ODE solver-based approaches in high-dimensional PBPK settings.
• Ablation studies showed multi-branch representation yields $>95\%$ lower MSE compared to single-branch models, and the inverse modeling phase achieves strictly lower parameter estimation error versus direct PINN approaches in all tested scenarios.

Implications and Future Directions

Theoretical Implications

MI-PINN introduces a structurally robust paradigm for inverse problems in dynamical systems by decoupling representation and parameter adaptation, inherently improving identifiability, optimization conditioning, and generalization across related tasks. The integration of clustering and closed-form analytic readouts further ensures model expressivity while stabilizing training.

Of particular note is the demonstrated capability of neural-symbolic hybridization in recovering missing mechanistic structures—a longstanding challenge in system identification with incomplete physical knowledge. This approach could generalize to broader classes of gray-box system learning problems in the natural sciences.

Practical Applications

The MI-PINN framework is directly applicable to fields requiring inference under severe data constraints: drug development, systems biology, engineering design, power systems modeling, among others. For PBPK specifically, the method's ability to recover kinetic parameters or entire dynamical components from limited clinical observations could significantly reduce the experimental burden in pharmacokinetic profiling and accelerate early-stage drug development workflows.

Prospects for Future Developments

Key directions for further research include:

• Uncertainty Quantification: Integrating calibrated Bayesian or ensemble-based UQ methods to provide confidence intervals on both inferred parameters and discovered mechanisms, crucial for clinical and regulatory acceptance.
• Extension to Stochastic and PDE-Governed Systems: Generalizing the framework to stochastic ODEs and partial differential equations, potentially leveraging operator-learning approaches for further scalability.
• Meta-Learned Transfer Across Physicochemical Contexts: Investigating more structured forms of meta-training to facilitate rapid adaptation to unseen systems with differing underlying physics.

Conclusion

The MI-PINN framework advances the state of the art in solving inverse problems for high-dimensional, coupled ODE systems under severe partial observability. By introducing a meta-learning-based, multi-branch physics-informed approach with closed-form pseudo-inverse updates, the method delivers reliable, efficient, and highly accurate parameter inference and mechanistic discovery. The empirical results, especially in PBPK settings, indicate substantial reductions in estimation error and computational cost relative to conventional and PINN-based baselines. The methods hold significant promise for scientific machine learning applications where mechanistic interpretability and sample efficiency are paramount (2605.03511).