A Physics-Guided Neural Framework for Rheology Measurement from Dynamical Laser Speckles

1. Problem Statement

Traditional rheometry (measuring storage modulus $G'(\omega)$ and loss modulus $G''(\omega)$) is invasive, time-consuming, and requires large sample volumes. While Laser Speckle Rheology (LSR) offers a non-contact, high-speed alternative, it faces significant hurdles in turbid (highly scattering) fluids:

• Multiple Scattering: In dense media, light undergoes multiple scattering events, distorting the intensity autocorrelation function $g_2(t)$ and invalidating simple single-scattering models.
• Ill-Posed Inversion: Recovering the Mean Square Displacement (MSD) and subsequently the viscoelastic moduli from $g_2(t)$ is an ill-posed inverse problem. Small errors in $g_2(t)$ are amplified during inversion, leading to unstable or physically ambiguous results.
• Parameter Dependency: Existing methods (like Diffusing Wave Spectroscopy) rely heavily on precise, sample-specific optical transport parameters (e.g., transport mean free path) which are difficult to measure in situ.
• Optical-Mechanical Coupling: Changes in particle size, concentration, or solvent properties alter the optical path, confounding the interpretation of the dynamic signal ($g_2(t)$) and making it difficult to isolate intrinsic Brownian dynamics.

2. Methodology

The authors propose a Physics-Guided Neural Network (PGNN) that bypasses the unstable direct inversion of $g_2(t)$. Instead of treating the problem as a black-box regression, they reformulate it as a physics-constrained parameter estimation problem.

A. Input Data

The framework utilizes two complementary data streams derived from the same speckle sequence:

1. Dynamic Signal: The intensity autocorrelation function $g_2(t)$, encoding time-dependent phase decorrelation (Brownian motion).
2. Static Signal: The speckle intensity histogram, encoding static optical properties (scattering path length diversity, detection geometry, and optical coefficients). This helps decouple optical effects from mechanical dynamics.

B. Network Architecture

The network follows a structured pipeline designed to embed physical laws:

1. Decay Analyzer: Processes raw $g_2(t)$ by generating four feature channels (original curve, first derivative, log-time gradient, log-time curvature) to extract multi-scale decay features.
2. Multi-Scale Mapping: Uses parallel convolutional branches with physically initialized kernels (based on exponential decay $e^{-t/\tau}$) to estimate initial relaxation times and amplitudes.
3. Attention Fusion: A cross-attention module fuses the dynamic features ($g_2(t)$) with static features (histogram) to correct for optical variations (concentration, particle size) without altering the intrinsic spectral shape.
4. Maxwell Spectrum Projection: The network outputs a Generalized Maxwell Relaxation Spectrum $H(\tau)$, represented as a set of discrete modes $\{G_i, \tau_i\}$. This step uses adaptive discretization and enforces physical constraints (non-negativity, smoothness, normalized weights).
5. Physics Forward Layer: A fixed, non-trainable layer maps the predicted $H(\tau)$ to the frequency-domain moduli $G'(\omega)$ and $G''(\omega)$ using the deterministic Generalized Maxwell equations (Stieltjes integral transforms). This ensures the output strictly satisfies causality and Kramers-Kronig relations.

C. Loss Function Design

The training objective combines data fidelity with physical constraints:

• Physics Loss ($L_{phy}$): Minimizes the log-domain error between predicted and reference $G'$, $G''$, and the loss tangent ($\tan \delta$). It also penalizes deviations in local power-law slopes.
• Reconstruction Loss ($L_{g2}$): Ensures the extracted modes can reconstruct the original $g_2(t)$ curve, preventing over-abstract feature learning.
• Regularization: Includes a separation constraint ($L_{scp}$) to prevent mode collapse and an entropy constraint ($L_{ent}$) to ensure a realistic distribution of relaxation times.

3. Key Contributions

• Physics-Guided Architecture: The first framework to directly infer a Maxwell relaxation spectrum from speckle data, embedding constitutive viscoelastic laws directly into the network's forward pass.
• Decoupling Optical and Mechanical Effects: By integrating the speckle intensity histogram as an auxiliary input, the model successfully distinguishes between changes in scattering conditions (optical) and changes in material rheology (mechanical).
• Robustness in Turbid Media: The method eliminates the need for in situ measurement of optical transport parameters, making it applicable to highly concentrated and polydisperse systems where traditional methods fail.
• Interpretability: Unlike black-box deep learning models, the output is a physically interpretable relaxation spectrum ($H(\tau)$), providing insight into the material's microstructural relaxation mechanisms.

4. Results

The framework was validated using polystyrene suspensions in glycerol-water mixtures with varying particle sizes (0.49–11 $\mu$m), concentrations (0.4–14% w/v), and solvent viscosities.

• Accuracy: The model achieved a Logarithmic RMSE ($RMSE_{log}$) as low as 0.009 for the loss modulus $G''(\omega)$ against reference rheometer data. The Median Absolute Percentage Error (MdAPE) for $G''$ was typically under 2.4% for seen conditions and under 15% for unseen conditions.
• Generalization:
  • Unseen Concentrations/Sizes: The model accurately predicted that relaxation times ($\tau$) depend on solvent viscosity (unchanged by concentration) while modulus magnitudes depend on particle loading.
  • Unseen Solvents: Even with solvent compositions outside the training set (e.g., G60W40, G80W20), the model maintained correct frequency dependence and scaling trends, despite higher magnitude errors.
• Ablation Study: Removing the histogram input led to systematic overestimation of moduli, confirming the histogram's critical role as an optical correction factor.
• Stability: Long-term testing (10 hours) showed minimal drift, with errors remaining within manageable limits, demonstrating suitability for continuous process monitoring.
• Efficiency: The use of histogram statistics rather than raw image sequences reduced computational costs by an order of magnitude compared to end-to-end image-based networks.

5. Significance

This work represents a paradigm shift in optical rheology. By moving away from unstable analytical inversions and toward physics-guided deep learning, the authors have created a "one-click" tool capable of:

1. Non-invasive, real-time characterization of complex fluids in industrial and biomedical settings.
2. Handling turbid media without the need for sample dilution or precise optical calibration.
3. Providing interpretable physical insights (relaxation spectra) rather than just numerical predictions.

The framework bridges the gap between data-driven AI and physical modeling, offering a robust solution for the rapid screening and in-line monitoring of viscoelastic materials in scenarios where traditional rheometers are impractical.