Jaber R. Mianroodi

Jaber R. Mianroodi, Nima H. Siboni, Dierk Raabe

Deep Reinforcement Learning (DRL) is employed to develop autonomously optimized and custom-designed heat-treatment processes that are both, microstructure-sensitive and energy efficient. Different from conventional supervised machine learning, DRL does not rely on static neural network training from data alone, but a learning agent autonomously develops optimal solutions, based on reward and penalty elements, with reduced or no supervision. In our approach, a temperature-dependent Allen-Cahn model for phase transformation is used as the environment for the DRL agent, serving as the model world in which it gains experience and takes autonomous decisions. The agent of the DRL algorithm is controlling the temperature of the system, as a model furnace for heat-treatment of alloys. Microstructure goals are defined for the agent based on the desired microstructure of the phases. After training, the agent can generate temperature-time profiles for a variety of initial microstructure states to reach the final desired microstructure state. The agent's performance and the physical meaning of the heat-treatment profiles generated are investigated in detail. In particular, the agent is capable of controlling the temperature to reach the desired microstructure starting from a variety of initial conditions. This capability of the agent in handling a variety of conditions paves the way for using such an approach also for recycling-oriented heat treatment process design where the initial composition can vary from batch to batch, due to impurity intrusion, and also for the design of energy-efficient heat treatments. For testing this hypothesis, an agent without penalty on the total consumed energy is compared with one that considers energy costs. The energy cost penalty is imposed as an additional criterion on the agent for finding the optimal temperature-time profile.

Iman Peivaste, Nima H. Siboni, Ghasem Alahyarizadeh et al.

Phase-field-based models have become common in material science, mechanics, physics, biology, chemistry, and engineering for the simulation of microstructure evolution. Yet, they suffer from the drawback of being computationally very costly when applied to large, complex systems. To reduce such computational costs, a Unet-based artificial neural network is developed as a surrogate model in the current work. Training input for this network is obtained from the results of the numerical solution of initial-boundary-value problems (IBVPs) based on the Fan-Chen model for grain microstructure evolution. In particular, about 250 different simulations with varying initial order parameters are carried out and 200 frames of the time evolution of the phase fields are stored for each simulation. The network is trained with 90% of this data, taking the $i$-th frame of a simulation, i.e. order parameter field, as input, and producing the $(i+1)$-th frame as the output. Evaluation of the network is carried out with a test dataset consisting of 2200 microstructures based on different configurations than originally used for training. The trained network is applied recursively on initial order parameters to calculate the time evolution of the phase fields. The results are compared to the ones obtained from the conventional numerical solution in terms of the errors in order parameters and the system's free energy. The resulting order parameter error averaged over all points and all simulation cases is 0.005 and the relative error in the total free energy in all simulation boxes does not exceed 1%.

Mohammad S. Khorrami, Pawan Goyal, Jaber R. Mianroodi et al.

The purpose of the current work is the development of a so-called physics-encoded Fourier neural operator (PeFNO) for surrogate modeling of the quasi-static equilibrium stress field in solids. Rather than accounting for constraints from physics in the loss function as done in the (now standard) physics-informed approach, the physics-encoded approach incorporates or "encodes" such constraints directly into the network or operator architecture. As a result, in contrast to the physics-informed approach in which only training is physically constrained, both training and output are physically constrained in the physics-encoded approach. For the current constraint of divergence-free stress, a novel encoding approach based on a stress potential is proposed. As a "proof-of-concept" example application of the proposed PeFNO, a heterogeneous polycrystalline material consisting of isotropic elastic grains subject to uniaxial extension is considered. Stress field data for training are obtained from the numerical solution of a corresponding boundary-value problem for quasi-static mechanical equilibrium. This data is also employed to train an analogous physics-guided FNO (PgFNO) and physics-informed FNO (PiFNO) for comparison. As confirmed by this comparison and as expected on the basis of their differences, the output of the trained PeFNO is significantly more accurate in satisfying mechanical equilibrium than the output of either the trained PgFNO or the trained PiFNO.

Mohammad S. Khorrami, Pawan Goyal, Soroush Motahari et al.

The purpose of the current work is the development of an approach to account for quasi-static mechanical equilibrium in empirical (i.e., data-based) models for the stress field employing neural approximations (NAs), which include neural networks (NNs) and neural operators (NOs), in particular Fourier NOs (FNOs). Rather than including such constraints from physics in the loss function as done in the (now standard) physics-informed approach, the current approach incorporates or "encodes" such constraints directly into the architecture of the NA. As a result, both NA training and output are physically constrained in the physics-encoded approach, in contrast to the physics-informed approach, in which only training is physically constrained. For the current constraint of divergence-free stress, a novel encoding approach based on a stress potential is proposed. As a "proof-of-concept" example application of the current approach, a physics-encoded FNO (PeFNO) is developed for a heterogeneous polycrystalline material consisting of isotropic elastic grains and subject to uniaxial extension. Stress field data for this purpose are obtained from the numerical solution of corresponding boundary-value problems for quasi-static mechanical equilibrium. For comparison with the PeFNO, this data is also employed to develop an analogous physics-guided FNO (PgFNO) and physics-informed FNO (PiFNO). As expected theoretically, and confirmed by this computational comparison, for comparable accuracy of the stress field itself as compared to the data, the stress field output by the trained and tested PeFNO is significantly more accurate in satisfying mechanical equilibrium than the output of either the PgFNO or the PiFNO.