⚡ electrical engineering

Learning Structural Eigenmodes with Modal Operator Network (ModalONet)

This paper introduces ModalONet, a novel neural operator framework that directly recovers a structural system's intrinsic modal properties—including mode shapes, natural frequencies, and damping ratios—from response fields alone by leveraging DeepONet's branch-trunk factorization as a learnable modal superposition, achieving high accuracy across various structural systems without the need for eigensolvers or labeled modal data.

Original authors: Saad Waheed, Shabbir Ahmed

Published 2026-08-03

📖 7 min read🧠 Deep dive

CC BY 4.0

Original authors: Saad Waheed, Shabbir Ahmed

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 you are listening to a symphony orchestra. To the untrained ear, it's just a beautiful, swirling wall of sound. But to a musician, that sound is actually a precise recipe: a specific set of notes (frequencies), played with specific volumes and durations (damping), by specific instruments (mode shapes) vibrating in perfect harmony. In the world of engineering, structures like bridges, airplane wings, and skyscrapers behave exactly like that orchestra. When wind or traffic hits them, they don't just shake randomly; they vibrate in hidden, mathematical patterns called "modes." Knowing these patterns is crucial. If a bridge starts vibrating in a new, weird pattern, it might mean a bolt is loose or the steel is fatigued. Engineers need to find these "notes" to keep us safe, but traditionally, figuring them out has been like trying to reverse-engineer a song just by listening to a muffled recording, often requiring complex math or expensive sensors everywhere.

Enter a new approach called ModalONet. Think of it as a super-smart detective that listens to a structure's "song" (its vibration data) and instantly figures out the sheet music without needing to know the instrument's blueprint first. While older AI tools were great at predicting how a bridge would shake if you pushed it, they weren't designed to tell you what the bridge's natural notes were. This paper introduces a method that flips the script: instead of predicting the future shake, it listens to the shake and extracts the hidden musical notes (frequency), the instrument's decay (damping), and the shape of the vibration (mode shape) directly from the noise. It's like teaching a computer to hear the individual instruments in a chaotic band and write down their exact scores, even if it's never seen the band before.

The Problem: Finding the Hidden Notes in the Noise

Imagine you have a guitar string. If you pluck it, it vibrates at a specific pitch. But real structures are messy. They are made of thousands of tiny parts, and when they vibrate, they do it in complex, overlapping ways. Sometimes, two different parts of a structure might vibrate at the exact same pitch (a "degenerate" mode), making it incredibly hard to tell them apart. Traditionally, engineers have two ways to find these hidden notes:

The Blueprint Method: They build a perfect computer model of the structure and solve a giant math puzzle to find the notes. But if the real bridge is slightly different from the blueprint (maybe the wind changed the material properties), the math puzzle gives the wrong answer.
The Listening Method: They measure the vibrations and use old-school signal processing to guess the notes. This works, but it's often a manual, tedious process where humans have to decide which "notes" are real and which are just background noise.

The authors of this paper asked a simple question: Can we teach a neural network (a type of AI) to listen to the vibrations and automatically figure out the hidden notes, without needing a blueprint or a human to guess?

The Solution: ModalONet's Musical Magic

The team created a tool called ModalONet. To understand how it works, imagine the AI is trying to reconstruct a song from a recording. Most AI models try to memorize the song. But ModalONet is built differently. It uses a special architecture (a "DeepONet") that is naturally structured like a musical score.

Think of the AI as having two parts working together:

The "Trunk" (The Shape): This part learns the shape of the vibration. It figures out how the bridge bends and twists in space. It's like learning the physical shape of the guitar string as it vibrates.
The "Branch" (The Sound): This part learns the timing of the vibration. It figures out how fast the sound fades away and what the pitch is. It's like learning the rhythm and the decay of the note.

The clever trick is that the authors realized this "Trunk + Branch" structure is mathematically identical to how vibrations actually work in the real world. So, when they train the AI to simply "reconstruct" the vibration data (make the AI's output match the real data), the AI is forced to naturally split the data into these physical shapes and sounds. It's like giving a child a pile of Lego bricks and asking them to build a house; if you tell them the house must have a roof and walls, they will naturally sort the bricks into those categories.

What They Found: A Near-Perfect Tune

The team tested ModalONet on four different "instruments":

A simple supported beam (like a diving board).
A cantilever beam (like a flagpole).
A rectangular plate (like a thin sheet of metal).
A square plate (a tricky case where two different vibration patterns share the exact same pitch).

They fed the AI noisy, simulated vibration data—just like a real sensor would record—and asked it to find the notes. The results were surprisingly accurate:

The Shapes: The AI recovered the vibration shapes with a "Modal Assurance Criterion" (a score of how close the shapes are) of 0.998 or higher. This means the AI's shapes were almost identical to the perfect mathematical shapes, even though it never saw the math.
The Pitch (Frequency): The AI guessed the natural frequencies with an error of less than 5%.
The Decay (Damping): This was the hardest part. The raw AI guess for how fast the vibration dies out was a bit off (sometimes by a lot). However, the authors added a simple "post-processing" step (a log-envelope regression) that acted like a fine-tuner. After this step, the error dropped to under 7%, with most modes being within 2% of the true value.

The Tricky Part: The "Twin" Problem

One of the biggest challenges the paper tackled was the "degenerate" case. Imagine a square drum. You can hit it so it vibrates left-to-right, or front-to-back. In a perfect square, these two patterns have the exact same pitch. To a normal computer, these two patterns look like a messy mix, and it might guess a weird, rotated pattern that doesn't make physical sense.

ModalONet solved this by forcing its "Trunk" to be "separable." This is a fancy way of saying it forces the AI to look for patterns that are simple products of X and Y, rather than a messy mix. This constraint prevented the AI from getting confused by the twins. It successfully identified that there were two distinct shapes sharing one pitch, rather than inventing a fake, rotated shape.

What This Means (and What It Doesn't)

The paper shows that we can use AI to "listen" to a structure and pull out its fundamental physical properties—how it moves, how fast it vibrates, and how it fades—without needing a perfect blueprint or a human expert to sort through the data.

However, the authors are careful to note the limits of their discovery.

It's a Simulation: All the tests were done on computer-generated data. The AI has never listened to a real, physical bridge in the wind yet.
It's Not a Universal Translator: Right now, the AI has to be trained separately for each specific structure. It can't just look at a new bridge and instantly know its notes; it needs to learn that specific bridge first.
Damping is Tricky: While the frequency and shape were found very accurately, the "damping" (how fast it stops) required that extra post-processing step to be truly accurate.

In short, ModalONet is a powerful new tool that turns the messy job of finding structural "notes" into a clean, automated process. It suggests that in the future, we might be able to attach a few sensors to a bridge, let this AI listen, and instantly get a full report on the structure's health, all without needing to know the bridge's engineering drawings. It's a step toward making our infrastructure "sing" its secrets to us, clearly and accurately.

Technical Summary: Learning Structural Eigenmodes with ModalONet

Problem Statement
Neural operators, such as DeepONet and the Laplace Neural Operator (LNO), have proven effective as surrogates for learning the forward response of dynamical systems (mapping excitation to response). However, they are typically trained to reproduce system outputs rather than recovering the system's intrinsic structural properties, specifically its modal basis (mode shapes, natural frequencies, and damping ratios). Traditional modal identification relies on solving eigenproblems of discretized finite-element models (requiring accurate geometry and material properties) or experimental modal analysis (involving user-dependent decisions, model-order selection, and discrete sensor data). While data-driven decomposition methods like Proper Orthogonal Decomposition (POD) and Dynamic Mode Decomposition (DMD) exist, their correspondence to physical modal parameters is often conditional on restrictive assumptions regarding mass, excitation, and linearity. Existing neural network approaches for modal identification often require fixed-length inputs, specific geometries, or rely on pre-processed spectral data, limiting their ability to learn continuous, mesh-free representations directly from raw time-domain response fields.

Methodology: ModalONet
The authors propose ModalONet, a neural operator architecture designed to recover the structural modal basis directly from measured response fields without eigensolvers or labeled modes. The method is grounded in the observation that the DeepONet branch–trunk factorization is mathematically equivalent to a learnable form of modal superposition.

Architecture:
- Trunk Network (Spatial): Maps spatial coordinates to continuous, mesh-free mode shapes $\phi_i(s)$ . For 2-D domains, a separable trunk is employed ( $\phi_i(x,y) = \psi_i(x)\chi_i(y)$ ) to mirror the product structure of analytical plate modes. This separability is crucial for resolving degenerate modes (modes sharing the same natural frequency, such as in a square plate) by preventing the network from learning arbitrary rotated mixtures of the degenerate subspace.
- Branch Network (Temporal): Uses a Laplace Neural Operator (LNO) specialized for free vibration. Instead of processing an input forcing function, the branch learns $r$ poles $\mu_i$ and residues $\beta_i$ . The temporal modal coordinates are reconstructed as $q_i(t) = \text{Re}\{\beta_i e^{\mu_i t}\}$ . This pole–residue parameterization allows natural frequencies ( $\omega_i = |\mu_i|$ ) and damping ratios ( $\zeta_i = -\text{Re}(\mu_i)/|\mu_i|$ ) to be read directly from the learned poles.
- Degenerate Mode Handling: For systems with degenerate frequencies (e.g., square plates), poles with initial frequencies coinciding within 2% are tied to a single shared frequency parameter, while damping remains distinct per mode.
Training Objective:
The network is trained in an unsupervised manner using only the raw response field $X(t, s)$ , avoiding spectral transforms or time-frequency preprocessing. The composite loss function $\mathcal{L}$ includes:
1. Reconstruction Loss ( $\mathcal{L}_{rec}$ ): Minimizes the difference between the reconstructed field and the measured data.
2. Orthonormality Constraints ( $\mathcal{L}_{orth}, \mathcal{L}_{norm}$ ): Enforces that the learned spatial modes are mutually orthogonal and normalized (Gram matrix approaches identity).
3. Temporal Projection Consistency ( $\mathcal{L}_{proj}$ ): Ensures each learned mode shape aligns with the data projected onto its specific temporal signature, normalized by temporal energy. This prevents fast-decaying or weakly excited modes from collapsing into dominant modes.
Post-Processing:
- Mode Assignment: Learned modes are ordered and aligned with analytical references using the Hungarian algorithm to maximize the Modal Assurance Criterion (MAC).
- Damping Refinement: Since the reconstruction loss weakly constrains decay rates (residues can absorb amplitude errors), a post-hoc SNR-weighted log-envelope regression is performed on the modal projections to refine damping ratio estimates.

Key Contributions

Operator Learning for Modal Identification: The work recasts modal identification as an operator learning problem, demonstrating that DeepONet factorization is a learnable modal superposition capable of recovering the full modal basis (shapes, frequencies, damping) from response data alone.
Unsupervised Training with Physical Constraints: The method trains solely on response fields using orthonormality and projection constraints, eliminating the need for labeled modes or eigensolvers. It introduces specific mechanisms (separable trunks, tied frequencies) to handle degenerate modes and energy-normalized projections to handle weak modes.
Continuous and Interpretable Representation: Unlike grid-bound methods, ModalONet yields continuous, mesh-free mode shapes. Every latent dimension corresponds to a physical mode, making the representation interpretable and verifiable against theory.

Results
The method was validated on four structural systems: simply supported and cantilever Euler–Bernoulli beams, and rectangular and square Kirchhoff plates.

Mode Shapes: Recovered with a Modal Assurance Criterion (MAC) of at least 0.998 for all modes, including the challenging non-sinusoidal cantilever shapes and the degenerate square plate modes (where the separable trunk successfully isolated the distinct $(2,1)$ and $(1,2)$ patterns).
Natural Frequencies: Recovered with relative errors within 5% (non-degenerate modes within 2%). Degenerate pairs shared a single tied frequency with an error of 4.9%.
Damping Ratios: Raw pole estimates were biased low (up to -97% error for degenerate pairs). However, the post-hoc log-envelope refinement reduced the worst-case error to 6.5%, with nine of twelve modes achieving errors below 2%.
Ablation Studies: Removing the temporal projection term caused fast-decaying modes to degrade significantly. Replacing the separable trunk with a fully connected one caused degenerate modes to collapse into rotated mixtures (MAC $\approx$ 0.5), confirming the necessity of both components.

Significance and Claims
The paper claims that ModalONet demonstrates the potential of neural operators as accurate and interpretable tools for modal identification. By recovering the modal basis directly from response fields, the method offers a data-driven alternative that does not rely on the forward solution or pre-defined operator structures. The authors emphasize that the learned representation is physically meaningful, with mode shapes that are continuous functions rather than discrete vectors, and parameters (frequency, damping) that are native to the network's architecture.

Limitations
The authors note that the study is limited to synthetic data generated under proportional damping assumptions. Practical challenges such as non-proportional damping, correlated noise, and model-order selection for unknown systems are not addressed. Furthermore, the current formulation trains independently for each structure; it does not yet function as a transferable operator across varying geometries or materials, which is identified as a direction for future research.