Physics-Informed Machine Learning in Biomedical Science and Engineering

TL;DR

The paper surveys physics-informed machine learning (PIML) for biomedical science and engineering, focusing on three core frameworks: physics-informed neural networks (PINNs), neural ordinary differential equations (NODEs), and neural operators (NOs). It explains how PINNs enforce governing equations within a neural loss to address forward and inverse problems, how NODEs provide continuous-time dynamics for physiological processes, and how NOs learn mappings between function spaces for rapid, operator-level predictions. The review covers representative method illustrations, broad biomedical applications (from CSF flow and soft tissue mechanics to PK/PD modeling and medical imaging), and current outlooks that emphasize uncertainty quantification, generalization, and integration with emerging AI technologies like foundation models and large language models. It concludes with a discussion of needs and directions, including multifidelity and multimodality data fusion, robust UQ, and scalable, interpretable PIML workflows to advance biomedical modeling and decision-making.

Abstract

Physics-informed machine learning (PIML) is emerging as a potentially transformative paradigm for modeling complex biomedical systems by integrating parameterized physical laws with data-driven methods. Here, we review three main classes of PIML frameworks: physics-informed neural networks (PINNs), neural ordinary differential equations (NODEs), and neural operators (NOs), highlighting their growing role in biomedical science and engineering. We begin with PINNs, which embed governing equations into deep learning models and have been successfully applied to biosolid and biofluid mechanics, mechanobiology, and medical imaging among other areas. We then review NODEs, which offer continuous-time modeling, especially suited to dynamic physiological systems, pharmacokinetics, and cell signaling. Finally, we discuss deep NOs as powerful tools for learning mappings between function spaces, enabling efficient simulations across multiscale and spatially heterogeneous biological domains. Throughout, we emphasize applications where physical interpretability, data scarcity, or system complexity make conventional black-box learning insufficient. We conclude by identifying open challenges and future directions for advancing PIML in biomedical science and engineering, including issues of uncertainty quantification, generalization, and integration of PIML and large language models.

Figures (5)

• Figure 1: Overview of Methods. (a) Standard neural network (NN) architecture. (b) Kolmogorov–Arnold Networks (KANs), which use separable inner functions $\phi_{ij}(x_j)$ and outer functions $\Phi_i$ for improved spectral representation. (c) Physics-Informed Neural Networks (PINNs) and Physics-Informed Kolmogorov–Arnold Networks (PIKANs) that embed governing physical laws, initial/boundary conditions, and observed data into a unified loss function. (d) NODEs or KAN-based ODEs used to model dynamic systems by solving $\dot{u} = f_\theta(u, t)$, incorporating data loss. (e) DeepONet and PI-DeepONet architectures, where branch and trunk network (e.g., NN or KAN) represent operator inputs and evaluation coordinates, respectively. (f) Fourier Neural Operator (FNO), Wavelet Neural Operator (WNO), and Laplace Neural Operator (LNO), which learn mappings between function spaces.
• Figure 2: CSF dynamics in the perivascular space of the brain of a live mouse. (a) Fluorescent tracers are imaged with two-photon microscopy for particle tracking and segmentation of the perivascular space (PVS). (b) Model inputs include sparse 2D velocities from Particle Tracking Velocimetry (PTV), the 3D domain geometry, and moving boundary conditions (MBCs) derived from segmentation. Collocation points are sampled and clustered with all data into ordered training groups. (c) The model assumes that the velocity is decomposed into a mean field plus Gaussian noise, $\boldsymbol{u} = \bar{\boldsymbol{u}} + \boldsymbol{\epsilon}_u$, where $\boldsymbol{\epsilon}_u \sim \mathcal{N}(0, \boldsymbol{\sigma}_u)$. Two NNs (with parameters $\boldsymbol{\theta}_{\bar{u}}$ and $\boldsymbol{\theta}_\sigma$) are trained to predict the mean fields ($\bar{\boldsymbol{u}}, p$) and the noise's standard deviation ($\boldsymbol{\sigma}_u$), respectively. Training minimizes a composite loss: a Negative Log-Likelihood (NLL) criterion for the noisy experimental data and boundaries and a mean-squared error for the governing PDEs. (d) The trained model outputs continuous fields for velocity, pressure, and uncertainty. From these, quantities of interest like volumetric flow rate ($Q$) and wall shear stress ($\tau$) are derived over the cardiac cycle ($T$). Figure adopted from toscano2024inferring.
• Figure 3: Integrated PK–PD model within the CMINNs framework for studying tumor response under multi-dose chemotherapy. (a) Drug concentration over time, obtained from the pharmacokinetics model in a repeated-dose regimen, is passed into a reduced PK–PD system of two ODEs. The rate of tumor cell death, $k_{1}(t)$, is modeled as a time-varying parameter: following the first dose, $k_{1}(t)$ exhibits a sharp spike corresponding to rapid death of highly sensitive tumor cells, while subsequent doses produce progressively smaller and delayed peaks, reflecting the emergence of drug resistance and pharmacokinetic tolerance. The efficacy index $k_{2}$ is represented as a piecewise constant parameter, decreasing across treatment cycles, indicating reduced overall drug potency. Together, these dynamics capture the adaptive survival of resistant and persistent cells within the tumor microenvironment. (b) Workflow for reducing the high-dimensional PK–PD system to two ODEs by introducing time-varying and piecewise constant parameters, which capture changes in parameter values without adding new compartments (ODEs) to the system. Using PINNs enables simultaneous inference of unknown parameters and system trajectories. This reduction provides interpretable insights into chemotherapy resistance, tolerance, and persistence dynamics. Figure adapted from the results of Model 4 in cminn. .
• Figure 4: Overview of CortexODE framework for cortical surface reconstruction. (a) End-to-end pipeline: brain MRI scans are segmented to extract white matter (WM), followed by connectivity filtering, distance transform, Gaussian smoothing, and topology correction to form an implicit surface. Marching cubes and mesh smoothing generate the initial surface. Two CortexODE modules are then applied: the first deforms the initial mesh into the WM surface (optimized with Chamfer loss) and the second expands the WM surface into the pial surface (optimized with vertex-wise MSE). (b) Signed distance function (SDF) and topology correction: WM segmentation is converted into a signed distance function $U(x) = s(x)\,d(x,\partial\Omega)$, where $s(x)=+1$ inside and $-1$ outside the WM region. Gaussian smoothing yields a continuous implicit representation, and topology correction ensures genus-0 surfaces suitable for reconstruction. (c) Neural ODE core: the deformation network $f_\theta$ receives vertex coordinates and local MRI features and outputs a velocity field. Vertex trajectories are evolved via $\tfrac{dx(t)}{dt} = f_\theta(x(t),V),\; x(0)=x_0$, producing diffeomorphic deformations to WM or pial surfaces while avoiding self-intersections. Figure adopted from cortexode2023.
• Figure 5: Biaxial mechanical testing data from a representative porcine tricuspid valve anterior leaflet (TVAL) tissue (a-b), and results from constitutive nonlocal operator models (PNO) trained on this dataset (c-e). (a) Demonstration of displacement field measurements via digital image correlation (DIC). (b) Displacements and forces of the tines in the biaxial test of all 1398 samples, collected from seven protocols (loading-unloading cycles). (c) Averaged relative l2-norm error of different models’ prediction for the displacement field given boundary conditions. Here, homoPNO denotes the constitutive nonlocal operator model with homogeneous collagen fiber orientation, and heterPNO is the constitutive model with heterogeneous fiber orientation. (d) Predicted first Piola-Kirchhoff stress field (in kPa) computed from DIC-measured displacements using the heterPNO model. (e) Prediction of collagen fiber orientation field by the heterPNO model versus the experimentally detected orientation. Figure adopted from jafarzadeh2025heterogeneous.