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Federated Scientific Machine Learning Framework for Multi-Scale Inverse Parameter Identification: Application to Nonlocal Strain Gradient Plasticity

This study introduces a novel inverse federated scientific machine learning (I-FedSciML) framework that enables the privacy-preserving, collaborative calibration of nonlocal strain gradient plasticity model parameters across diverse micro-scale experimental datasets without centralizing raw data, achieving high predictive accuracy even under significant noise conditions.

Original authors: George Z. Voyiadjis, Babur Deliktas, Aiman Tariq

Published 2026-08-25

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: George Z. Voyiadjis, Babur Deliktas, Aiman Tariq

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

At the scale of a human hair or a grain of sand, the rules that govern how metal bends and stretches begin to change. In the everyday world, a thick steel beam and a thin steel wire of the same material should behave similarly when bent; their strength is a fixed property of the metal itself. But when scientists shrink the size of a metal sample down to the microscopic level, they find something unexpected: the smaller the piece, the stronger it becomes. A tiny wire resists twisting far more than a thick one, and a microscopic beam resists bending with a force that classical physics cannot explain. This phenomenon, known as size-dependent plasticity, happens because the internal structure of the metal—specifically the tiny defects called dislocations that allow it to deform—accumulates differently when the sample is so small. To predict how these tiny structures will behave, engineers use complex mathematical models that include a special "length scale" to account for this extra strength. However, figuring out the exact numbers for these models has been a stubborn problem. The data needed to calibrate them comes from delicate experiments performed in different labs, often on different materials, and the results are sparse and scattered. Traditionally, researchers have had to analyze each experiment in isolation, which often leads to uncertain or conflicting answers about what the material is actually doing.

A team of researchers has now proposed a new way to solve this puzzle by bringing these scattered experiments together without ever moving the raw data. They developed a system called inverse federated scientific machine learning, a method that allows multiple computers to learn from each other while keeping their private information safe. Imagine a group of scientists in different countries, each holding a unique set of experimental results on a piece of metal. Instead of sending their sensitive data to a central server where it could be lost or stolen, they send only the "lessons" they learned from their own data to a central coordinator. The coordinator mixes these lessons to create a smarter, shared understanding, which is then sent back to the scientists to refine their local models. In this study, the researchers applied this approach to the problem of identifying the hidden parameters of a nonlocal strain gradient plasticity model. They treated each distinct experiment—whether it was a thin nickel beam, a copper wire, or an aluminum foil—as an independent "client" in a digital network. The system learned to map the specific geometry of each sample, such as its thickness and grain size, directly to the specific numbers needed to make the physics model work for that exact piece of metal.

The researchers tested this framework using real-world data from four different types of microscopic experiments: bending thin annealed nickel beams, bending LIGA nickel foils, bending aluminum beams, and twisting thin copper wires. In each case, the system successfully identified a unique set of material parameters for every single experiment, even though the data came from different sources with different sizes and shapes. For the nickel beams, the model learned that the strength of the material depended heavily on how thick the beam was, accurately reproducing the experimental curves with a high degree of precision. The same was true for the copper wires; the system correctly predicted how much force was needed to twist wires of different diameters, capturing the subtle differences in how they hardened under stress. The results were remarkably accurate, with the model's predictions matching the experimental measurements so closely that the statistical error was often less than one percent. This suggests that the system can effectively learn the underlying physics of how size changes material behavior, even when the data is messy or comes from very different setups.

To ensure the method was robust, the researchers also tested how well it handled noise, simulating the kind of small measurement errors that happen in real laboratories. They introduced artificial errors ranging from one percent to ten percent into the data to see if the system would break. Even with ten percent noise, which is a significant amount of error, the framework remained stable. While the predictions became slightly less precise and the identified numbers showed a bit more variation, the system still captured the main trends of how the materials behaved. It did not collapse into nonsense; instead, it provided a reliable estimate of the material's properties despite the interference. This resilience is crucial for real-world application, where experimental data is rarely perfect. The study demonstrates that it is possible to build a collaborative, privacy-preserving network that learns from diverse, distributed experiments to solve complex physical problems. By keeping the data local and sharing only the learned insights, this approach offers a new path for scientists to calibrate advanced material models without compromising the security or ownership of their experimental results. The work does not claim to have solved every problem in material science, but it provides a powerful new tool for understanding how the microscopic world behaves, one that respects the boundaries between different research groups while uniting their knowledge.

Technical Summary: Federated Scientific Machine Learning Framework for Multi-Scale Inverse Parameter Identification

Problem Statement
Classical plasticity theories fail to describe size-dependent plastic deformation observed in micro-scale experiments (e.g., micro-bending, micro-torsion) because they lack an intrinsic material length scale. Strain Gradient Plasticity (SGP) models address this by incorporating strain gradients and geometrically necessary dislocations (GNDs). However, calibrating nonlocal SGP models remains challenging. The effective material length scale is often not a fixed constant but depends on microstructural features (grain size), specimen size, and deformation modes. Consequently, the model response relies on several interdependent parameters (reference stress, hardening modulus, hardening exponent, and evolving length-scale relations).

Traditional inverse identification methods (e.g., Finite Element Model Updating, Virtual Fields Method) typically treat each experimental dataset independently. This approach fails to leverage information across heterogeneous datasets obtained from different laboratories, geometries, or loading conditions. Furthermore, the scarcity of micro-scale data and the potential need to protect proprietary or sensitive experimental data across distributed research groups necessitate a framework that can perform collaborative calibration without centralizing raw data.

Methodology
The authors propose an Inverse Federated Scientific Machine Learning (I-FedSciML) framework designed to identify parameters for a generalized nonlocal SGP model with a varying length scale. The methodology integrates three core components:

Physics-Based Forward Model: The study utilizes a generalized SGP formulation (Voyiadjis and Abu Al-Rub, 2005) where the effective plastic strain combines local and gradient-dependent terms. The material length scale is formulated as a function of specimen size, grain size, and plastic strain. Analytical and numerical solutions are derived for size-dependent micro-bending (thin beams) and micro-torsion (thin wires) to generate differentiable forward models ( $\mathcal{F}_{SGP}$ ) capable of predicting moment–curvature and torque–twist responses.
Geometry-Conditioned Inverse Neural Network: Instead of using neural networks as direct black-box surrogates for material response, the framework employs a shared neural network as an inverse operator. This network maps normalized geometric descriptors (grain size $d$ , specimen size $D$ , and thickness/width $h$ ) to a client-specific SGP parameter vector ( $\sigma_{ref}, \sigma_0, m$ ). The network outputs are constrained via bounded sigmoid transformations to ensure physical admissibility.
Federated Learning Protocol: The framework treats each experimental curve (representing a unique geometry and loading condition) as an independent "client."
- Local Training: Each client performs local optimization to minimize the discrepancy between the predicted response (generated by the forward model using the network's predicted parameters) and the local experimental data.
- Global Aggregation: A central server aggregates the local model updates using the Federated Averaging (FedAvg) algorithm. This allows the shared inverse network to learn a global mapping between geometry and parameters without ever accessing the raw experimental data of individual clients.

Key Contributions

First Application of Federated SciML to SGP: This work presents the first inverse federated scientific machine learning framework specifically for identifying parameters of size-dependent nonlocal SGP models.
Privacy-Preserving Collaborative Calibration: The framework enables the collaborative calibration of constitutive models across distributed datasets (different labs, geometries, materials) while preserving data privacy by keeping raw experimental curves local to each client.
Geometry-Conditioned Parameter Identification: The method successfully identifies distinct, client-specific parameter sets that account for the specific geometric and microstructural characteristics of each experiment, rather than forcing a single global parameter set on heterogeneous data.
Differentiable Physics Integration: The approach embeds the SGP forward model directly into the training loop, ensuring that the identified parameters are physically consistent with the governing differential equations of strain gradient plasticity.

Results
The framework was validated using four benchmark micro-scale experimental datasets:

Annealed Nickel Beams (Micro-bending): Three clients with varying thicknesses and grain sizes. The framework achieved $R^2$ values between 0.982 and 0.990. The identified parameters showed that the hardening modulus ( $\sigma_0$ ) was the primary driver for distinguishing responses, while the reference stress ( $\sigma_{ref}$ ) converged to lower bounds.
LIGA Nickel Foils (Micro-bending): Four clients with varying characteristic sizes. The framework achieved $R^2$ values between 0.925 and 0.974. In this case, $\sigma_{ref}$ showed significant client-dependent variation, indicating its importance in differentiating these specific responses.
Aluminum Beams (Micro-bending): Two clients. The framework achieved $R^2$ values of 0.898 and 0.904. The hardening modulus again played a dominant role in fitting the data.
Copper Wires (Micro-torsion): Five clients with varying wire diameters. The framework achieved high accuracy with $R^2$ values ranging from 0.985 to 0.997.

Noise Sensitivity Analysis:
The study evaluated robustness by introducing synthetic Gaussian noise (1%, 2.5%, 5%, and 10%) into the copper torsion data.

Parameter Bias and Uncertainty: As noise levels increased, parameter bias and standard deviation increased. At 10% noise, bias in $\sigma_{ref}$ reached nearly 18% for some clients, and prediction uncertainty widened.
Predictive Capability: Despite increased noise, the framework maintained reliable predictive capability. Even at 10% noise, the mean predictions preserved the main size-dependent trends, and the 95% prediction intervals remained physically meaningful, though wider.

Significance and Claims
The authors claim that the I-FedSciML framework provides an effective and privacy-preserving route for the collaborative calibration of nonlocal size-dependent constitutive models. The study demonstrates that:

Federated learning can successfully handle the heterogeneity of micro-scale experimental data (different geometries, materials, and loading modes) without centralizing raw data.
The geometry-conditioned inverse operator can recover distinct parameter sets for different clients while learning a unified global relationship between geometry and material behavior.
The framework is robust enough to handle realistic measurement noise, maintaining the ability to capture size-dependent strengthening effects even under significant data perturbation.

The work establishes a foundation for future collaborative constitutive modeling across distributed laboratories, addressing the limitations of independent calibration and the barriers to data sharing in scientific research. Future extensions suggested by the authors include incorporating richer physical descriptors (e.g., texture, strain-rate effects) and generalizing the approach to other nonlocal models such as coupled damage-plasticity and gradient-enhanced fracture.

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