💻 computer science

An Explainable Physics-Informed Neural Frequency-Response Framework for Shunt-Parameter Identification in Semi-Active Piezoelectric Tuned Mass Dampers

This paper proposes an explainable, physics-informed neural framework that leverages synthetic frequency-response data to enable fast, robust, and interpretable identification of hidden structural and shunt parameters in semi-active piezoelectric tuned mass dampers.

Original authors: Andreas Georgiou, Vasileios Gkatsis, Vasileios Sioros, George Giannakopoulos, Nikolaos Chrysochoidis, Christoforos Rekatsinas

Published 2026-08-10

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CC BY 4.0

Original authors: Andreas Georgiou, Vasileios Gkatsis, Vasileios Sioros, George Giannakopoulos, Nikolaos Chrysochoidis, Christoforos Rekatsinas

Original paper licensed under CC BY 4.0 (http://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 world where giant bridges, skyscrapers, and wind turbines are constantly swaying in the wind or shaking from traffic. To stop them from wobbling too much, engineers attach heavy weights called "tuned mass dampers" that swing in the opposite direction to cancel out the motion. Think of it like a counterweight on a seesaw. However, these traditional weights are like old-fashioned radios: they are tuned to one specific frequency. If the wind changes speed or the building settles slightly, the "radio" gets out of tune, and the damper stops working.

To fix this, scientists created "semi-active" versions that can change their tune on the fly. They do this by using piezoelectric materials—special crystals that turn mechanical shaking into electricity. By connecting these crystals to an external electrical circuit (a "shunt"), engineers can tweak the circuit's resistance and inductance to change how the damper behaves, much like turning a dial to find a new radio station. But here's the tricky part: figuring out exactly which dial settings to use for a specific building is incredibly hard. It usually requires complex math, massive amounts of data, or slow, trial-and-error testing. If you want to know the right settings for a new building, you might have to run a slow computer simulation for every single possibility, which is like trying to find a needle in a haystack by checking every single piece of hay one by one.

This paper introduces a clever new tool called PI-NFRF (Physics-Informed Neural Frequency-Response Framework) that acts like a super-smart detective for these vibrating systems. Instead of guessing or running slow simulations every time, the researchers built a "neural network"—a type of artificial intelligence—that learned to look at the "sound" of a vibrating structure (its frequency response) and instantly guess the correct electrical settings (resistance and inductance) needed to calm it down.

The secret sauce is that this AI isn't just a black box that memorizes answers. The researchers taught it using a "physics-informed" approach. They built a digital twin of the damper based on real laws of physics and used it to generate millions of fake but realistic vibration patterns. They then trained the AI to work backward: given a vibration pattern, what electrical settings created it? Crucially, the AI doesn't just guess; it checks its own work. If it predicts a setting, it runs that setting back through the physics model to see if it recreates the original vibration. If the physics model says "no, that doesn't match," the AI learns to try again. This ensures the AI's guesses are always physically possible and consistent with reality.

The results are impressive. When the researchers tested this AI on real-world experimental data it had never seen before, it was incredibly accurate. For the electrical resistance, it was off by an average of only 0.41%, and for the inductance, it was off by just 0.66%. In plain terms, the system successfully identified the correct settings for various test cases, such as predicting a resistance of 21.999981 Ω when the target was 22.0 Ω, or 60.1483 Ω when the target was 60.0 Ω. This means the system can identify the right settings in a split second, without needing to solve a new, slow math problem for every single measurement.

But the researchers didn't stop at just making a fast calculator; they wanted to know how the AI was thinking. They used "explainable AI" techniques to peek inside the brain of the neural network. They found that the AI organizes its knowledge in a smooth, logical way, grouping similar vibration patterns together. They also discovered that the AI pays the most attention to specific "sweet spots" in the sound spectrum—specifically around 50–52 Hz and 103–110 Hz—where the vibration changes most dramatically when the electrical settings are tweaked. Finally, they used a technique called "symbolic distillation" to boil down the AI's complex logic into a simple, readable math formula that describes how the vibration changes based on the settings.

In short, this paper suggests that we can now tune these advanced vibration dampers much faster and more reliably than before. By combining the speed of AI with the reliability of physics, the researchers created a system that not only solves the problem quickly but also explains why it found the solution, giving engineers the confidence to trust the AI's recommendations. While the current tests were done on specific setups and simulated data, the method shows great promise for making our buildings and structures safer and more stable without the need for endless, slow calculations.

Technical Summary: An Explainable Physics-Informed Neural Frequency-Response Framework for Shunt-Parameter Identification in Semi-Active Piezoelectric Tuned Mass Dampers

Problem Statement

Semi-active shunted piezoelectric tuned mass dampers (SATMDs) offer broadband vibration mitigation by combining a resonant auxiliary mass with a piezoelectric transducer and a reconfigurable resistive-inductive (RL) shunt circuit. While effective, the performance of SATMDs is highly sensitive to structural detuning and operating conditions. A critical challenge in their practical deployment is the rapid and robust identification of shunt parameters (resistance $R$ and inductance $L$ ) from measured frequency response functions (FRFs).

Conventional approaches rely on analytical tuning rules or numerical optimization based on prescribed electromechanical models. These methods often require repeated optimization for every new measurement, are computationally demanding, and can be unreliable when discrepancies exist between the nominal model and the actual system. Furthermore, purely data-driven machine learning models often lack physical consistency, require large labeled experimental datasets, and suffer from poor interpretability, which is a barrier to trust in high-stakes vibration control applications.

Methodology

The paper proposes the Physics-Informed Neural Frequency-Response Framework (PI-NFRF), a frequency-domain approach that integrates a differentiable physics-based solver with a neural inverse model and explainability techniques. The framework operates through three primary use cases:

1. Differentiable Forward Physics Model

The core of the framework is a differentiable electromechanical solver derived from the coupled dynamics of the SATMD. The system is modeled as a three-degree-of-freedom electromechanical system (primary mass, auxiliary mass, and electrical charge) governed by a complex dynamic stiffness matrix $\mathbf{Z}(\omega; \boldsymbol{\theta})$ .

Formulation: The equilibrium is defined as $\mathbf{Z}(\omega; \boldsymbol{\theta})\mathbf{x}(\omega; \boldsymbol{\theta}) = \mathbf{f}(\omega)$ , where $\boldsymbol{\theta}$ includes structural parameters ( $m_2, k_p$ ) and shunt parameters ( $R, L$ ).
Differentiability: The solver computes complex-valued acceleration FRFs directly from the parameters using a differentiable linear solver, allowing gradients to flow back through the physics model for optimization and training.

2. Inverse Learning (Use Case II)

Instead of solving an optimization problem for each new measurement, a neural network is trained to perform amortized inference, mapping complex acceleration FRFs directly to shunt parameters ( $R, L$ ).

Data Generation: A synthetic dataset is generated by sampling the forward solver across a range of $R$ and $L$ values. Controlled noise is added only to the input FRFs during the inverse training stage, while the forward physics model remains noise-free to preserve physical consistency.
Physics-Consistent Training: The network is not trained solely on parameter labels. Instead, it employs a closed-loop loss function: predicted parameters are fed back into the forward solver, and the loss is computed based on the mismatch between the reconstructed FRF and the target FRF. This ensures the learned inverse map is constrained by the governing electromechanical physics.
Architecture: A fully connected feed-forward network (256-128-64 hidden layers) maps the real and imaginary parts of the acceleration FRFs to normalized $R$ and $L$ values.

3. Explainability and Analysis (Use Case III)

To address the "black box" nature of neural networks, the framework employs three interpretability strategies:

Latent Space Organization: Principal Component Analysis (PCA) is applied to the 64-dimensional latent vectors of the inverse network to visualize how the model organizes different electromechanical tuning states.
Frequency-Wise Sensitivity: Local sensitivities ( $\partial \hat{R}/\partial \mathbf{X}$ and $\partial \hat{L}/\partial \mathbf{X}$ ) are computed to identify which spectral regions (frequencies) are most influential for parameter inference.
Symbolic Distillation: A reduced scalar descriptor ( $\xi_{FRF}$ ) representing the global deviation of the FRF from a nominal reference is derived. Symbolic regression (using PySR) is then applied to find a compact analytical expression linking this descriptor to the physical parameters ( $R, L, m_2, k_p$ ).

Key Results

Parameter Identification (Use Case I)

The framework successfully identified physical parameters ( $m_2, k_p, R, L$ ) by minimizing the reconstruction error of an experimental FRF.

Accuracy: The identified parameters closely matched reference values, with a relative FRF reconstruction error of $2.35 \times 10^{-6}$ .
Interpretability: The method recovered physically meaningful corrections to nominal values (e.g., a 72.7% adjustment to absorber mass and 122% to piezoelectric stiffness), demonstrating that the differentiable solver can calibrate complex models against experimental data.

Blind Inverse Tuning (Use Case II)

The inverse neural network was evaluated on six blind experimental configurations not seen during training.

Performance: The model achieved mean absolute errors of 0.0823 $\Omega$ for resistance and $5.72 \times 10^{-4}$ H for inductance.
Relative Error: Mean relative errors were 0.41% for $R$ and 0.66% for $L$ .
Generalization: The low errors across the tested range (from low to high $R$ and $L$ ) indicate successful synthetic-to-experimental transfer and the ability to interpolate within the sampled parameter space without re-optimization.

Explainability Findings (Use Case III)

Latent Structure: PCA projections revealed a smooth, curved manifold in the latent space, indicating the network learns a structured representation of the electromechanical tuning. However, because $R$ and $L$ varied jointly in the blind cases, the model learned a combined tuning coordinate rather than fully disentangled axes.
Spectral Sensitivity: Sensitivity analysis identified two critical frequency bands: 50–52 Hz and 103–110 Hz. These regions correspond to resonance peaks, anti-resonance shifts, and modal interactions, confirming the network relies on physically informative spectral features.
Symbolic Distillation: A compact symbolic expression was derived that explained approximately 86% of the variance in the global FRF descriptor. The expression highlighted the coupled nature of the system, showing that mechanical and electrical parameters cannot be treated as independent contributors to the response.

Significance and Claims

The paper claims that PI-NFRF provides a data-efficient and explainable alternative to traditional optimization and pure data-driven approaches for SATMD tuning. Its primary contributions are:

Physics-Informed Inference: By embedding the governing electromechanical equations into the training loop, the framework ensures that parameter predictions are physically consistent, allowing for rapid inference without solving a new optimization problem for each measurement.
Synthetic-to-Experimental Transfer: The framework demonstrates that training on synthetic data generated by a differentiable physics solver can effectively generalize to real-world experimental FRFs, reducing the need for large labeled experimental datasets.
Interpretability: Unlike opaque neural networks, PI-NFRF offers engineering insight through latent space analysis, frequency-wise attribution (identifying critical spectral regions), and symbolic distillation (providing compact analytical relations).
Modest Scope: The authors explicitly state that the results validate the framework's ability to interpolate within the sampled parameter space. They do not claim global identifiability of parameters or that the symbolic distillation recovers the exact governing equations; rather, it provides an engineering-level summary of the dominant parameter dependencies.

The work positions PI-NFRF as a step toward trustworthy, real-time retuning of semi-active electromechanical absorbers, grounded in validated analytical formulations while leveraging the speed of neural inference.