Quantification of total uncertainty in the physics-informed reconstruction of CVSim-6 physiology

TL;DR

The paper tackles the challenge of quantifying total uncertainty in physics-informed reconstructions of physiology by introducing MC X-TFC, a Monte Carlo extension of the eXtreme Theory of Functional Connections for gray-box ODE identification. It demonstrates how physics-informed regularization interacts with aleatoric, epistemic, and model-form uncertainty, first with a pedagogical harmonic ODE and then with the CVSim-6 lumped-parameter cardiovascular model. The results show robust state and parameter estimation under sparse and noisy data, and reveal how the choice of discrepancy terms influences model-form uncertainty, with inertia-like regularization (inductance) mitigating bias in pulmonary flows. The approach enables fast, online uncertainty quantification without offline training, offering a practical pathway for data-physics fusion in physiological digital twins and ICU monitoring contexts.

Abstract

When predicting physical phenomena through simulation, quantification of the total uncertainty due to multiple sources is as crucial as making sure the underlying numerical model is accurate. Possible sources include irreducible aleatoric uncertainty due to noise in the data, epistemic uncertainty induced by insufficient data or inadequate parameterization, and model-form uncertainty related to the use of misspecified model equations. Physics-based regularization interacts in nontrivial ways with aleatoric, epistemic and model-form uncertainty and their combination, and a better understanding of this interaction is needed to improve the predictive performance of physics-informed digital twins that operate under real conditions. With a specific focus on biological and physiological models, this study investigates the decomposition of total uncertainty in the estimation of states and parameters of a differential system simulated with MC X-TFC, a new physics-informed approach for uncertainty quantification based on random projections and Monte-Carlo sampling. MC X-TFC is applied to a six-compartment stiff ODE system, the CVSim-6 model, developed in the context of human physiology. The system is analyzed by progressively removing data while estimating an increasing number of parameters and by investigating total uncertainty under model-form misspecification of non-linear resistance in the pulmonary compartment. In particular, we focus on the interaction between the formulation of the discrepancy term and quantification of model-form uncertainty, and show how additional physics can help in the estimation process. The method demonstrates robustness and efficiency in estimating unknown states and parameters, even with limited, sparse, and noisy data. It also offers great flexibility in integrating data with physics for improved estimation, even in cases of model misspecification.

Figures (15)

• Figure 1: CVSim-6 model circuit and default output.
• Figure 2: Schematic of the X-TFC algorithm for performing gray-box identification of the pulmonary flux discrepancy. Input weights and biases are randomly selected. The last step solves iteratively a least-squares problem.
• Figure 3: Decomposition of total uncertainty (epistemic and aleatoric) in the reconstruction of a harmonic ODE solution from noisy data using MC X-TFC. $B$ denotes the bound of the uniform distribution $\mathcal{U}[-B, B]$ from which the input weights and biases of the hidden layer are randomly initialized. For comparison, results from ensemble PINN and B-PINN are also reported. The estimated values of $k$ are presented in Table \ref{['tab:example_0']}.
• Figure 4: Reconstructed $x(t)$ and epistemic uncertainty computed using MC X-TFC for a varying degree of physics-informed regularization, where $\lambda_1$ in \ref{['eq:example_0_loss']} denotes the penalty coefficient in the loss function. Both the error in the predicted mean and the predicted uncertainty grow significantly as a result of increasingly relying on (missing) data as $\lambda_1$ is reduced.
• Figure 5: Quantification of epistemic uncertainty in the reconstructed solution of a harmonic equation from an inconsistent dataset of varying size. The first row presents an estimate for the solution $x(t)$ while the second row represents an estimate for the discrepancy $\delta(t)$. The predicted (epistemic) uncertainty of $\delta(t)$ is used to characterize model-form uncertainty. On the leftmost plots, we present the result obtained by ignoring $\delta(t)$, and using the misspecified differential equation, hence demonstrating the use of MC X-TFC in the context of model misspecification.

Theorems & Definitions (2)

• Remark 1
• Remark 2