1. Introduction

The hybrid rheological model based on a second-order NARX (Nonlinear AutoRegressive with eXogenous inputs) neural network developed in this work solves the fundamental problem in solid mechanics — accurate prediction of flow stress in metallic materials during high-temperature plastic deformation while accounting for complex loading history. Unlike traditional phenomenological models unable to adequately describe hardening-softening competition, and physically-based models requiring detailed microstructural data, the proposed approach ensures correct history accounting without explicit internal variable parameterization. The model demonstrates unprecedented accuracy (average relative error 1.5%) when verified on 7075/2.5% TiC composite in the 300–500°C range.

2. Development of the NARX Neural Network Model

The key achievement is the justification of a physically correct input parameter set parameterizing the internal material state using macroscopic quantities from the loading history only. The two-step autoregressive memory is formed by the pair of current and previous flow stresses, while exogenous inputs include kinematic characteristics (strain rates of current and previous steps), time increments, and process temperature.

Accumulated strain is deliberately excluded from inputs, as it loses informativeness during active structural softening (dynamic recrystallization, dislocation recovery) when it fails to reflect the actual microstructure state.

3. NARX Network Training Methodology

An original training methodology has been developed based on NARX transformation into an equivalent multilayer perceptron (MLP), where experimental stress-strain pairs replace feedback connections. Data preparation includes:

• cubic spline smoothing to eliminate sensor noise;

• variable time step $\Delta t$ and non-monotonic strain-rate paths $\dot{\varepsilon}(t)$ for enhanced generalization;

• random shuffling of input-output pairs.

This strategy eliminates sensitivity to fixed time steps characteristic of LSTM/GRU, ensuring stable training for sequences of any length.

4. Material Characterization and Experimental Validation

For verification, a model system was selected — metal matrix composite 7075 (Al-5.8%Zn-2.2%Mg-1.7%Cu) + 2.5 vol.% TiC (mean particle size 1.2 μm), synthesized by liquid-phase stir casting with homogenization annealing. The composite exhibits typical complex rheology: dislocation pinning by particles, dynamic recovery, recrystallization, and damage accumulation.

Uniaxial compression tests covered wide strain-rate ranges with variable loading paths not used in training. The hybrid model showed exceptional accuracy: average relative error 1.5%, all predictions within ±5% experimental scatter. Local curve features — hardening peaks and softening troughs — were perfectly reproduced, confirming robust history-dependence capture.

5. Model Advantages and Applications

The proposed model surpasses existing approaches by key characteristics:

• Computational efficiency: MLP speed while retaining memory;

• Overfitting resistance: physically-motivated parameter selection;

• Discretization invariance: variable-step training;

• Universality: applicable to steels, alloys, composites with hardening-softening competition.

Unlike differential structure evolution models requiring explicit microstructural equations, NARX adaptively identifies hidden patterns from macroscopic stress response. The model integrates into finite element packages for hot forming optimization, enabling accurate prediction of complex industrial loading trajectories.

Future developments include extension to non-isothermal regimes, multiaxial states, and damage incorporation.

6. Conclusions

A novel hybrid NARX-based rheological model has been successfully developed and validated for predicting flow stress evolution under high-temperature deformation with complex loading histories. Key innovations — physically-justified input parameterization, variable-step training methodology, exclusion of uninformative strain accumulation — ensure 1.5% accuracy and generalization capability. This work establishes a new paradigm in solid mechanics constitutive modeling, providing industry-ready tools for metal forming process optimization.