🔬 condensed matter

Direct inference of viscoelastic memory from chirp rheometry via physics-informed Gaussian processes

This paper introduces a physics-informed Gaussian process framework that directly infers viscoelastic memory kernels from raw time-domain chirp rheometry data, bypassing traditional frequency-domain reduction to enable rapid, retraining-free prediction of material responses to unseen deformation histories and the resolution of material evolution within a single experiment.

Original authors: Isaac Y. Miranda-Valdez, Juha Koivisto, Mikko J. Alava

Published 2026-08-18

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CC BY 4.0

Original authors: Isaac Y. Miranda-Valdez, Juha Koivisto, Mikko J. Alava

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

Soft materials, from the gels in our food to the tissues in our bodies, possess a unique kind of memory. Unlike a simple spring that snaps back immediately when released, these substances remember how they were deformed in the past. The stress they feel at any given moment is not determined solely by their current shape, but by the entire history of how they were stretched or squeezed before. This hereditary behavior is the key to predicting how such materials will perform under real-world conditions, yet measuring it has traditionally been a slow and tedious process. Scientists usually probe these materials by shaking them at one specific frequency, waiting for the signal to settle, and then moving to the next frequency. To map out the full behavior of a material, this process can take hours, a timeframe that is often too long for materials that are changing or aging during the measurement. Furthermore, the standard way of analyzing the data involves compressing thousands of raw measurements into a few summary numbers, a step that inevitably discards a vast amount of information and can introduce errors.

A team of researchers at Aalto University in Finland has developed a new way to bypass these limitations, allowing them to read a material's memory directly from a single, rapid experiment. Instead of shaking the material at one frequency at a time, they use a "chirp" signal, a strain that sweeps continuously through a wide range of frequencies in just a few seconds. This method generates a dense stream of data, capturing thousands of stress and strain pairs in the time it used to take to gather a handful of points. The researchers then applied a sophisticated statistical framework, known as a physics-informed Gaussian process, to this raw data. Rather than first converting the data into frequency summaries, this approach uses the fundamental laws of physics to interpret the time-domain record directly. It treats the material's memory as a set of building blocks, testing different mathematical shapes to see which one best explains the observed stress. By doing so, the method identifies the material's specific memory rules and quantifies the uncertainty in those rules without ever needing to discard the raw data or assume a specific model beforehand.

The researchers tested this approach on several different soft materials, each presenting a unique challenge. In one experiment, they analyzed a solution of wormlike micelles, a type of soft matter that typically behaves like a simple fluid with a single relaxation time. The standard method would require a long frequency sweep to confirm this, but the new framework analyzed a chirp lasting only two seconds. The data was sufficient to pinpoint the material's single relaxation time and confirm that it behaves almost exactly like a classic fluid, with only a tiny, measurable deviation that the method could detect with high precision. The researchers also demonstrated that the model learned from this short burst of data was not just memorizing the specific signal used. When they applied the trained model to a completely different, longer chirp with a different starting phase and frequency range, it accurately predicted the material's response, including behavior at frequencies the original short experiment never even touched. This proved that the system had learned the true physical law governing the material, not just a pattern specific to the test signal.

The power of the method became even more apparent when dealing with noisy or complex materials. In a study of a cellulose nanofiber hydrogel, the data was significantly noisier than in the micellar experiment, a situation where traditional analysis often fails. The new framework successfully extracted a clear, scale-free memory pattern from the noise, matching the results of a conventional measurement that took sixty times longer. In another test involving a photopolymer resin, the material exhibited two distinct behaviors acting simultaneously: a permanent elastic network and a dissipative, fluid-like fraction. The method was able to separate these two contributions within a single time trace, quantifying how much stress came from each mechanism and providing a clear picture of the material's internal structure. Finally, the researchers used the technique to watch a material change in real time. By analyzing a UV-curing acrylate, they detected a violation of the material's time-invariance, observing the viscosity rise steadily as the material hardened during the single chirp experiment. This allowed them to track the onset of chemical mutation directly in the time domain, without needing a pre-defined model of how the curing process should behave.

By treating the raw time-domain record as the primary source of truth, this approach transforms how scientists understand soft matter. It replaces the slow, step-by-step process of spectral estimation and model fitting with a single, unified inference problem. The result is a method that is faster, more robust against noise, and capable of revealing material properties that were previously hidden or required prohibitively long measurements to find. It offers a way to see the memory of soft materials with a clarity that was previously out of reach, turning seconds of experimental data into a detailed map of how a material remembers its past.

Technical Summary: Direct Inference of Viscoelastic Memory from Chirp Rheometry via Physics-Informed Gaussian Processes

Problem Statement
Soft materials exhibit viscoelastic behavior where stress is determined by a weighted integral over the entire deformation history, rather than the instantaneous strain. Identifying this "memory" is critical for predicting material behavior under real-world loading. The standard rheological approach, Small Amplitude Oscillatory Shear (SAOS), is reproducible but suffers from two practical limitations: (1) full frequency sweeps are time-intensive (minutes to hours), making them unsuitable for evolving systems like gelling or aging materials; and (2) constitutive inference occurs only after spectral estimation (Fourier transformation), meaning noise and finite-window artifacts introduced during signal reduction propagate irreversibly into the fitted model.

Chirp rheometry offers a solution by compressing hours of measurement into seconds, generating dense time-domain data ( $10^3$ – $10^4$ stress-strain pairs). However, the standard analysis pipeline for chirp data mimics SAOS: the dense time series is windowed and Fourier-transformed to estimate storage ( $G'$ ) and loss ( $G''$ ) moduli before fitting a constitutive model. This process discards the majority of the measured information, creating an "inferential bottleneck" where biases in spectral estimation limit the accuracy of the recovered constitutive law. Furthermore, existing machine learning approaches in rheology often lack generalizability; models trained on specific waveforms frequently fail to predict responses to different deformation protocols because they learn waveform-specific mappings rather than intrinsic material properties.

Methodology
The authors propose a physics-informed Gaussian Process (GP) framework that performs direct probabilistic constitutive inference in the time domain, bypassing intermediate spectral estimation.

Physics-Informed Feature Construction: The framework embeds the Boltzmann superposition principle directly into the GP input space. Instead of using raw time coordinates, the input features are constructed as causal convolutions of the measured strain-rate history ( $\dot{\gamma}$ ) with candidate memory kernels ( $\phi_i$ ).
$\sigma(t) = \sum_{i=1}^n c_i x_i(t; \kappa)$
where $x_i(t) = \int_0^t \phi_i(t-t'; \kappa) \dot{\gamma}(t') dt'$ . Here, $\kappa$ represents shape parameters (e.g., relaxation times, fractional exponents) and $c_i$ are scalar prefactors (e.g., moduli, viscosities).
Gaussian Process Regression: A GP is assigned over the map from these engineered features to the measured stress.
- Linear Kernel: For materials in the linear viscoelastic regime, a linear kernel is used. This makes the GP equivalent to Bayesian linear regression in the feature space, where the posterior sensitivities ( $\partial \bar{\sigma} / \partial x_i$ ) directly yield the constitutive prefactors ( $c_i$ ).
- RBF Kernel: For cases where linearity or time-translation invariance cannot be assumed (e.g., mutating materials), a Radial Basis Function (RBF) kernel allows for a nonlinear stress-feature map. The time-resolved sensitivity of the GP prediction to the features serves as a diagnostic for material evolution.
Model Selection and Training: The framework trains jointly on all parameters (shape parameters $\kappa$ and GP hyperparameters) by maximizing the Evidence Lower Bound (ELBO). Model selection among candidate constitutive laws (e.g., Maxwell, Fractional Maxwell, Scott Blair, Fractional Kelvin-Voigt) is performed using Akaike (AIC) and Bayesian (BIC) information criteria, calculated directly from the raw time-domain data.

Key Results
The framework was validated across five distinct experimental scenarios:

Maxwellian Memory in Wormlike Micelles: Applied to a 2-second chirp on a wormlike micellar solution, the method selected a Fractional Maxwell Liquid (FML) model over classical Maxwell and other fractional variants. It recovered a relaxation time ( $\tau_c \approx 1.49$ s) and a fractional exponent ( $\beta \approx 0.014$ ) that places the material near its Maxwell limit. Crucially, the model inferred a characteristic frequency ( $\omega_c \approx 0.67$ rad/s) that lay outside the excited frequency band of the training chirp, demonstrating the ability to extrapolate beyond the training window.
Protocol Independence: A GP trained on a 2-second chirp was tested against an independent 14-second chirp with a different bandwidth, duration, and initial phase. The trained GP predicted the stress response with high accuracy (RMSE comparable to the training fit) and correctly resolved the terminal crossover frequency, confirming that the method learns a constitutive law rather than a waveform-specific mapping.
Scale-Free Memory in Critical Gels: On a noisy 7-second chirp of a cellulose nanofiber hydrogel, the method recovered a Scott Blair (critical gel) memory kernel. It achieved a coefficient of determination ( $R^2 = 0.863$ ) and identified the fractional exponent ( $\alpha \approx 0.063$ ) and quasi-property ( $V$ ) consistent with a 451-second discrete frequency sweep, despite the chirp data being two orders of magnitude shorter and significantly noisier.
Stress Apportionment in Photopolymer Resins: For a photopolymer resin modeled by a Fractional Kelvin-Voigt (FKV) model, the framework successfully decomposed the stress response into two parallel memory features (a quasi-elastic network and a power-law dissipative branch). It quantified the uncertainty in this decomposition and resolved a weak power-law relaxation in the network branch that standard Fourier analysis failed to capture due to noise.
Detection of Material Mutation: Applied to a UV-curing acrylate system, the framework used an RBF kernel to detect time-translation invariance violations. The time-resolved sensitivity of the GP prediction to the memory feature (interpreted as dynamic viscosity) showed a monotonic increase of +8.1% over a single chirp, directly capturing the onset of material mutation (crosslinking) without a prescribed kinetic model. A control measurement on the same material at a later curing stage showed no such drift.

Significance and Claims
The paper claims to resolve the limitations of standard chirp analysis by replacing the sequence of spectral estimation followed by constitutive fitting with a single physics-informed probabilistic inference problem posed directly on raw time-domain data.

Direct Inference: The method infers the memory kernel (a constitutive property) rather than the specific waveform, enabling predictions on unobserved deformation histories without retraining.
Robustness to Noise and Short Duration: By utilizing the full density of time-resolved data, the framework recovers constitutive parameters from short or noisy records where conventional Fourier-based methods fail or produce sparse, artifact-laden estimates.
Uncertainty Quantification: The probabilistic formulation provides quantified uncertainty for all inferred parameters (moduli, exponents, relaxation times) and allows for variance decomposition between constitutive features and observation noise.
Model-Free Diagnostics: The time-resolved sensitivity serves as a diagnostic tool for detecting non-stationarity (material evolution) and non-linearities, offering a protocol-independent fingerprint of structural changes.

The authors conclude that this approach establishes physics-informed feature construction as a route to generalizable machine learning in rheology, capable of handling model selection, uncertainty-aware parameter estimation, and evidence-based hypothesis testing directly from the output of a single short experiment.

Technical Summary: Direct Inference of Viscoelastic Memory from Chirp Rheometry via Physics-Informed Gaussian Processes

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