⚡ electrical engineering

Sensor placement study for estimation of Reynolds number in flow past a circular cylinder using an Inverse Physics-Informed Neural Network

This study utilizes an Inverse Physics-Informed Neural Network (I-PINN) to demonstrate that sensor placement is more critical than sensor quantity for accurately estimating the Reynolds number in flow past a circular cylinder, revealing that wake zones yield the most reliable results while upstream zones are ineffective.

Original authors: Dinesh Sai Palli, Vardhan Nagulla, Sai Saketha Chandra Athkuri

Published 2026-11-03

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CC BY 4.0

Original authors: Dinesh Sai Palli, Vardhan Nagulla, Sai Saketha Chandra Athkuri

Original paper licensed under CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine a river flowing past a bridge pier. The water does not simply slide around the obstacle; it swirls, stretches, and sometimes breaks into a chaotic wake of eddies behind the structure. The intensity and pattern of this behavior depend on a single, invisible number known as the Reynolds number. This value acts as a master switch for fluid motion, determining whether the flow remains smooth and orderly or becomes turbulent and violent. In the real world, engineers often need to know this number to design safe ships, bridges, and offshore platforms, yet they rarely have the luxury of measuring the entire flow field. Instead, they must rely on a handful of sensors placed at specific points to guess the hidden parameter that governs the whole system. The challenge is finding the right spots to place those sensors so that a few measurements can reveal the truth about the entire flow.

A team of researchers at the National Institute of Technology Calicut in India tackled this problem by simulating the flow of water past a circular cylinder, a classic shape used to study fluid dynamics. They focused on a specific scenario where the flow is steady and predictable, avoiding the chaotic turbulence that would make the task nearly impossible. Using a powerful new type of computer program called a physics-informed neural network, they created a digital twin of the flow. This program is unique because it does not just learn from data; it is also forced to obey the fundamental laws of physics, specifically the equations that describe how fluids move and how pressure changes. By training this digital model first, the researchers generated a perfect, detailed map of the flow, including the speed of the water and the pressure at every point. They then used this perfect map to create a set of synthetic sensor readings, mimicking what real instruments would record if they were placed in the actual flow.

The core of the study was a systematic search for the best place to put these virtual sensors. The researchers divided the area around the cylinder into five distinct zones: the region immediately behind the cylinder, the middle of the wake, the far wake, the area in front of the cylinder where the water is still undisturbed, and the thin layers of swirling fluid that separate the wake from the main flow. They tested each zone by feeding the inverse version of their neural network a small number of sensor readings, ranging from ten to fifty points, and asked the computer to work backward to find the Reynolds number. The goal was to see if the computer could correctly identify the hidden value based solely on the data from that specific zone.

The results revealed a clear and surprising truth: where you place the sensor matters far more than how many sensors you use. The researchers found that placing sensors in the wake behind the cylinder yielded the most accurate results. In the near wake, right behind the cylinder, and in the far wake, the computer could estimate the Reynolds number with an error of roughly ten to fifteen percent. The middle wake performed similarly well, though with slightly more variation. In contrast, sensors placed in the thin shear layers surrounding the wake were less effective, producing errors around thirty percent. The worst location by a significant margin was the upstream region, the area in front of the cylinder where the water flows smoothly and uniformly. Even with fifty sensors in this zone, the computer failed to guess the Reynolds number accurately, with errors exceeding forty-six percent.

The study also explored why these differences exist. The researchers discovered that the accuracy of the estimate depends on how much the local flow is influenced by the fluid's internal friction, or viscosity. In the wake behind the cylinder, the fluid moves slowly and swirls, meaning the friction forces are strong and play a major role in the motion. This makes the flow highly sensitive to the Reynolds number, allowing the sensors to pick up a clear signal. In the upstream region, however, the water moves fast and smoothly, dominated by momentum rather than friction. Here, the Reynolds number has very little effect on the local flow, so the sensors cannot provide enough information for the computer to make a good guess. The number of sensors did not fix this problem; adding more sensors to a poor location did not improve the accuracy, confirming that the quality of the information depends entirely on the physical nature of the flow at that spot.

The researchers also tested how sensitive the method was to the starting guess. They found that if the computer began with a guess that was too far from the true value, it would get stuck and fail to find the correct answer, no matter how good the sensor data was. This suggests that while the method is powerful, it still requires a reasonable starting point to work effectively. Ultimately, the study demonstrates that for engineers trying to estimate flow properties from limited data, the strategy should be to place sensors where the fluid is most active and where friction plays a visible role, rather than simply scattering more instruments across the field. The location of the measurement is the key to unlocking the hidden physics of the flow.

Technical Summary: Sensor Placement Study for Reynolds Number Estimation in Flow Past a Circular Cylinder

Problem Statement
Determining unknown flow parameters, such as the Reynolds number ( $Re$ ), from limited experimental measurements is a critical challenge in engineering. Traditional approaches often rely on optimization loops embedding Computational Fluid Dynamics (CFD) solvers, which require solving adjoint problems for every parameter update. This process is computationally expensive and requires fully differentiable solvers. While Physics-Informed Neural Networks (PINNs) offer a mesh-free alternative by treating unknown parameters as trainable weights, their effectiveness in recovering global parameters from sparse sensor data depends heavily on where those sensors are placed. This study addresses the gap in understanding how sensor location influences the recovery of a single global parameter ( $Re$ ) using an Inverse PINN (I-PINN), specifically for flow past a circular cylinder.

Methodology
The study employs a two-stage framework using a steady, two-dimensional, incompressible flow at $Re = 40$ (a regime with a steady, symmetric wake and no vortex shedding to ensure training stability).

Forward PINN Validation: A forward PINN was first trained to solve the Navier-Stokes equations for the flow field $(u, v, p)$ . The network architecture consisted of 8 hidden layers with 20 neurons each. The model was validated against classical experimental data (Apelt [22]) for surface pressure coefficients ( $C_p$ ), confirming its ability to reproduce the correct physics, including symmetric recirculation bubbles and stagnation points.
Synthetic Data Generation: The validated forward solution served as the ground truth to generate synthetic sensor data.
Inverse PINN Formulation: An I-PINN was developed to recover the unknown kinematic viscosity ( $\nu = 1/Re$ ) from sparse observations. The parameter was introduced as a trainable scalar, $\log(1/Re)$ , to ensure stable gradient updates. The total loss function combined data loss (Mean Squared Error between predictions and sparse sensor readings), PDE residuals, and boundary condition losses.
Optimization Strategy: Training followed a three-phase schedule:
- Warm-up: 1,000 epochs with $Re$ fixed at an initial guess ( $Re_0 = 20$ ) to establish a reasonable flow field.
- Joint Training: 6,000 epochs using Adam optimizers to update network weights and $Re$ simultaneously.
- Fine-tuning: Up to 2,000 iterations using L-BFGS-B on a flattened vector of all parameters.
Experimental Design: The study systematically varied sensor placement across five distinct physical zones: Near Wake (Z1), Mid Wake (Z2), Far Wake (Z3), Upstream Region (Z4), and Shear Layers (Z5). For each zone, the number of sensors ( $N$ ) was swept from 10 to 50. The study was repeated in three independent runs to assess sensitivity to initialization and sampling.

Key Contributions and Results

Sensor Location Dominates Count: The primary finding is that sensor location is significantly more critical than sensor count. Increasing the number of sensors ( $N$ ) from 10 to 50 did not monotonically reduce the error in any zone. In some cases, error even increased with more sensors.
Zone Performance Ranking:
- Wake Zones (Z1, Z2, Z3): These zones yielded the most accurate estimates. The Near Wake (Z1) achieved the lowest error (9.9% at $N=10$ in Run 1), followed by the Far Wake (Z3) and Mid Wake (Z2). Errors in these zones generally ranged between 9.9% and 21.6%.
- Shear Layers (Z5): Performance was intermediate, with errors consistently between 28.2% and 31.7%.
- Upstream Zone (Z4): This zone was consistently the least effective, with errors ranging from 46.3% to 62.3%. The flow here is nearly uniform, providing little sensitivity to the Reynolds number.
Physical Mechanism (Viscous Fraction): The study identified the "viscous fraction" ( $F$ $F$ ), defined as the ratio of the viscous term to the convective term in the momentum equation, as the key predictor of zone informativeness.
- Zones with high $F$ (Wake regions where viscous diffusion drives the wake recovery) correlated strongly with low recovery errors (Spearman $\rho = -0.80$ ).
- Zones with low $F$ (Upstream region where convection dominates) resulted in poor recovery.
- Gradient energy ( $G$ ) alone was insufficient to explain the ranking, as shear layers had high gradients but low $F$ and poor recovery.
Sensitivity to Initial Guess: The recovery of $Re$ is highly sensitive to the initial guess ( $Re_0$ ). Starting with $Re_0 = 1$ (Stokes regime) or $Re_0 = 100$ caused the optimizer to stall or diverge, respectively. Successful recovery required an initial guess within the same order of magnitude as the true value.
Optimizer Behavior: While the L-BFGS-B phase significantly reduced the total loss (by an order of magnitude), it had minimal effect on the recovered $Re$ value. This suggests that the accuracy limit is determined by the information content of the sensor data and the conditioning of the inverse problem, rather than the optimization algorithm's ability to descend the loss landscape.

Significance and Claims
The paper claims that for inverse PINN applications aimed at recovering global flow parameters, sensor placement must be guided by the physical sensitivity of the flow field to that parameter, specifically the relative magnitude of viscous terms in the momentum balance. The study demonstrates that simply increasing sensor density is ineffective if the sensors are placed in regions where the parameter of interest (Reynolds number) has little influence on the local flow physics.

The authors conclude that while I-PINNs can recover $Re$ with reasonable accuracy (best case ~10% error) using sparse data, achieving high absolute accuracy (e.g., <5%) remains challenging with the current setup. The work provides practical guidance: sensors should be placed in regions where the viscous term carries a sizeable share of the momentum balance (e.g., recirculation regions or recovering wakes), and a physically informed initial guess is essential for convergence. The study does not claim to solve the general inverse problem for all flow regimes but establishes a baseline for steady, laminar flows where PINNs are known to train reliably.

Technical Summary: Sensor Placement Study for Reynolds Number Estimation in Flow Past a Circular Cylinder

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