Physics-informed neural networks for hidden boundary detection and flow field reconstruction

TL;DR

This work tackles the inverse problem of detecting hidden solid boundaries and reconstructing surrounding flow from sparse measurements by introducing a physics-informed neural network (PINN) that embeds a body fraction field $\phi$ to distinguish fluid and solid domains. By penalizing NS and Euler dynamics with $\phi$-dependent terms and enforcing appropriate boundary conditions, the method simultaneously recovers boundary geometry, boundary motion (for moving bodies), and full flow fields under both incompressible and compressible regimes. The framework is validated on three benchmarks—steady flow past a fixed cylinder, in-line oscillating cylinder, and subsonic flow over a NACA 0012 airfoil—showing accurate boundary inference, robust flow reconstruction, and reliable estimation of boundary kinematics under sparse and noisy data. These results demonstrate the method’s potential for real-world flow diagnostics and control where direct boundary observations are unavailable, with implications for aero/hydrodynamics, biomedical flow analysis, and underwater acoustics. The approach also provides a foundation for extending PINNs to more complex geometries, 3D flows, and multi-physics scenarios with limited measurements.

Abstract

Simultaneously detecting hidden solid boundaries and reconstructing flow fields from sparse observations poses a significant inverse challenge in fluid mechanics. This study presents a physics-informed neural network (PINN) framework designed to infer the presence, shape, and motion of static or moving solid boundaries within a flow field. By integrating a body fraction parameter into the governing equations, the model enforces no-slip/no-penetration boundary conditions in solid regions while preserving conservation laws of fluid dynamics. Using partial flow field data, the method simultaneously reconstructs the unknown flow field and infers the body fraction distribution, thereby revealing solid boundaries. The framework is validated across diverse scenarios, including incompressible Navier-Stokes and compressible Euler flows, such as steady flow past a fixed cylinder, an inline oscillating cylinder, and subsonic flow over an airfoil. The results demonstrate accurate detection of hidden boundaries, reconstruction of missing flow data, and estimation of trajectories and velocities of a moving body. Further analysis examines the effects of data sparsity, velocity-only measurements, and noise on inference accuracy. The proposed method exhibits robustness and versatility, highlighting its potential for applications when only limited experimental or numerical data are available.

Figures (17)

• Figure 1: Schematic of PINNs for inferring the shape of a hidden stationary boundary. For the NS equations, the network models the flow field by predicting velocity $(u, v)$, pressure $p$, and body fraction $\phi$, taking spatial coordinates $(x, y)$ as inputs. For the Euler equations, the only difference is that the network also outputs density $\rho$. The loss functions comprise two components: a physical equation part and a known labeled data part.
• Figure 2: Schematic of PINNs for inferring the shape, velocity, and trajectory of a hidden moving boundary. The network architecture consists of two components: a primary network and a sub-network. The primary network models the flow field by predicting velocity $(u, v)$, pressure $p$, and body fraction $\phi$, taking spatial coordinates $(x, y)$ and time $t$ as inputs. The sub-network captures the velocity of the moving boundary, using time $t$ as input and outputting the boundary velocity components $U$ and $V$. Both networks optimized simultaneously. The loss functions comprise two components: a physical equation part and a known labeled data part.
• Figure 3: Predicted body fraction $\phi$ distributions and inferred boundary shapes for the steady flow around a fixed cylinder at three data resolutions (100, 500, and 1000 randomly sampled points). The first row shows the predicted body fraction distributions, while the second row presents the comparison between the predicted and the exact cylinder boundary shapes, along with the corresponding relative $L^2$ errors.
• Figure 4: Comparison between the predicted (trained with 100 random data points) and the reference flow fields for the velocity $u$, $v$, and pressure $p$ in the steady flow around a fixed cylinder. The bottom row depicts the point-wise absolute errors.
• Figure 5: The forward and inverse problem setups, along with the distribution of training points, for the in-line oscillating cylinder in fluid. (a) Computational domain and prescribed motion of the oscillating cylinder for the forward problem, following the configuration in Ref. zhu2024physics. (b) Computational domain for the inverse problem (where $\Omega$ represents the whole region, $\Omega_1$ denotes the region with available velocity measurements, and $\Omega_2$ is the target reconstruction region containing a hidden moving boundary) with spatial distribution of measurement data points ($10 \times 10$) in $\Omega_1$ and 30 extra sensor measurement points uniformly placed along the boundary of $\Omega_2$. (c) Spatial-temporal distribution of randomly sampled collocation points (in $\Omega$ and $\Omega_2$), used for enforcing physical constraints.