Ioannis Stefanou

Filippo Masi, Ioannis Stefanou

Data-driven and deep learning approaches have demonstrated to have the potential of replacing classical constitutive models for complex materials. Yet, the necessity of structuring constitutive models with an incremental formulation has given rise to data-driven approaches where physical quantities, e.g. deformation, blend with artificial, non-physical ones, such as the increments in deformation and time. Neural networks and the consequent constitutive models depend, thus, on the particular incremental formulation, fail in identifying material representations locally in time, and suffer from poor generalization. Herein, we propose a new approach which allows, for the first time, to decouple the material representation from the incremental formulation. Inspired by the Thermodynamics-based Artificial Neural Networks (TANN) and the theory of the internal variables, the evolution TANN (eTANN) are continuous-time and, therefore, independent of the aforementioned artificial quantities. Key feature of the proposed approach is the identification of the evolution equations of the internal variables in the form of ordinary differential equations, rather than in an incremental discrete-time form. In this work, we focus attention to juxtapose and show how the various general notions of solid mechanics are implemented in eTANN. The capabilities as well as the scalability of the proposed approach are demonstrated through several applications involving a broad spectrum of complex material behaviors, from plasticity to damage and viscosity (and combination of them). Finally, we show that the proposed approach can be used to speed-up multiscale analyses, by virtue of asymptotic homogenization. eTANN provide excellent results compared to detailed fine-scale simulations and offer the possibility not only to describe the average macroscopic material behavior, but also micromechanical, complex mechanisms.

David Michael Riley, Guglielmo Scovazzi, Ioannis Stefanou

Porous media containing cracks, fractures, or internal discontinuities arise throughout subsurface geomechanics, biomechanics, and materials science. Numerical simulation of the coupled hydromechanical response is inherently challenging because the pressure and displacement fields are tightly coupled through the Biot equations, requiring stable mixed formulations. These difficulties are compounded when cracks are present, because standard mesh-conforming approaches require costly, labor-intensive, body-fitted meshing, while unfitted methods often require cut-cell integration, enrichment functions, or additional stabilization. In this work, we use an alternative approach, we adapt the shifted interface method to coupled transient poroelasticity with embedded interfaces. The method replaces the true crack by a surrogate approximation where interface conditions are transferred through local expansions. A unified derivation yields shifted forms for both hydraulic transmission and mechanical traction coupling. Two enforcement strategies are extensively compared: a weak (integral) enforcement and a strong (pointwise) enforcement. Four test cases of increasing geometric complexity (offset mesh-aligned, boundary-intersecting angled, embedded angled, and multi-crack configurations) validate the formulation. Away from crack tips, interface residuals converge as O(h); near tips, localized post-processing artifacts degrade the global rate, but first-order convergence is recovered when a small tip region is excluded. A multi-crack demonstration with four simultaneously embedded cracks of distinct geometry and interface properties confirms the practical applicability of the framework. These results support the shifted interface method as a practical framework for poroelastic crack modeling on non-body-fitted meshes with geometrically complex embedded interfaces.

Diego Gutierrez-Oribio, Alexandros Stathas, Ioannis Stefanou

Deep Geothermal Energy, Carbon Capture and Storage, and Hydrogen Storage hold considerable promise for meeting the energy sector's large-scale requirements and reducing CO$_2$ emissions. However, the injection of fluids into the Earth's crust, essential for these activities, can induce or trigger earthquakes. In this paper, we highlight a new approach based on Reinforcement Learning for the control of human-induced seismicity in the highly complex environment of an underground reservoir. This complex system poses significant challenges in the control design due to parameter uncertainties and unmodeled dynamics. We show that the reinforcement learning algorithm can interact efficiently with a robust controller, by choosing the controller parameters in real-time, reducing human-induced seismicity and allowing the consideration of further production objectives, \textit{e.g.}, minimal control power. Simulations are presented for a simplified underground reservoir under various energy demand scenarios, demonstrating the reliability and effectiveness of the proposed control-reinforcement learning approach.

Diego Gutiérrez-Oribio, Ioannis Stefanou

Induced seismicity caused by fluid extraction or injection in underground reservoirs is a major challenge for safe energy production and storage. This paper presents a robust output-feedback controller for induced seismicity mitigation in geological reservoirs described by a coupled 3D PDE-ODE model. The controller is nonlinear and robust (MIMO Super-Twisting design), producing a continuous control signal and requiring minimal model information, while accommodating parameter uncertainties and spatial heterogeneity. Two operational outputs are regulated simultaneously: regional pressures and seismicity rates computed over reservoir sub-regions. Closed-loop properties are established via explicit bounds on the solution and its time derivative for both the infinite-dimensional dynamics and the nonlinear ODE system, yielding finite-time or exponential convergence of the tracking errors. The method is evaluated on the Groningen gas-field case study in two scenarios: gas production while not exceeding the intrinsic seismicity of the region, and combined production with CO$_2$ injection toward net-zero carbon operation. Simulations demonstrate accurate tracking of pressure and seismicity targets across regions under significant parameter uncertainty, supporting safer reservoir operation while preserving production objectives.

Giovanni Piunno, Ioannis Stefanou, Cristina Jommi

This paper introduces a novel approach that combines Proper Orthogonal Decomposition (POD) with Thermodynamics-based Artificial Neural Networks (TANN) to capture the macroscopic behavior of complex inelastic systems and derive macroelements in geomechanics. The methodology leverages POD to extract macroscopic Internal State Variables from microscopic state information, thereby enriching the macroscopic state description used to train an energy potential network within the TANN framework. The thermodynamic consistency provided by TANN, combined with the hierarchical nature of POD, allows to reproduce complex, non-linear inelastic material behaviors as well as macroscopic geomechanical systems responses. The approach is validated through applications of increasing complexity, demonstrating its capability to reproduce high-fidelity simulation data. The applications proposed include the homogenization of continuous inelastic representative unit cells and the derivation of a macroelement for a geotechnical system involving a monopile in a clay layer subjected to horizontal loading. Eventually, the projection operators directly obtained via POD, are exploit to easily reconstruct the microscopic fields. The results indicate that the POD-TANN approach not only offers accuracy in reproducing the studied constitutive responses, but also reduces computational costs, making it a practical tool for the multiscale modeling of heterogeneous inelastic geomechanical systems.

Diego Gutiérrez-Oribio, Ioannis Stefanou

Induced seismicity associated with gas production poses major operational and societal challenges, as illustrated by the Groningen field in the Netherlands. While many studies have focused on forecasting seismicity under prescribed production scenarios, fewer works address the inverse problem: designing operational strategies that minimize seismicity while maintaining production objectives. In this paper, we propose a control-oriented methodology for operating Groningen under induced-seismicity mitigation constraints. We employ a cascade model coupling pore-pressure diffusion with seismicity rate (SR) dynamics, and complement it with a stochastic event-generation procedure to convert the continuous SR field into a synthetic earthquake catalog with event times, locations, and magnitudes. From this catalog, we estimate regional SR measurements and design a robust feedback controller that computes well-rate commands to regulate the SR toward a desired reference while satisfying operational requirements, including prescribed production constraints. The proposed control architecture explicitly accounts for injection and extraction flux limits (actuator saturation). The well fluxes generated by the controller are updated at discrete-time intervals (digital control). We validate the modeling components against Groningen data and illustrate the approach through numerical experiments under different scenarios, including various control update periods and gain selections, as well as combined production with compensating injection (e.g., reinjection of nitrogen). The results illustrate how the proposed framework can reduce seismicity levels in a controlled manner while maximizing production targets.

Filippo Masi, Ioannis Stefanou

The mechanical behavior of inelastic materials with microstructure is very complex and hard to grasp with heuristic, empirical constitutive models. For this purpose, multiscale, homogenization approaches are often used for performing reliable, accurate predictions of the macroscopic mechanical behavior of solids and structures. Nevertheless, the calculation cost of such approaches is extremely high and prohibitive for real-scale applications involving inelastic materials. Here, we propose the so-called Thermodynamics-based Artificial Neural Networks (TANN) for the constitutive modeling of materials with inelastic and complex microstructure. Our approach integrates thermodynamics-aware dimensionality reduction techniques and thermodynamics-based deep neural networks to identify, in an autonomous way, the constitutive laws and discover the internal state variables of complex inelastic materials. The efficiency and accuracy of TANN in predicting the average and local stress-strain response, the free-energy and the dissipation rate is demonstrated for both regular and perturbed two- and three-dimensional lattice microstructures in inelasticity. TANN manage to identify the internal state variables that characterize the inelastic deformation of the complex microstructural fields. These internal state variables are then used to reconstruct the microdeformation fields of the microstructure at a given state. Finally, a double-scale homogenization scheme (FEMxTANN) is used to solve a large scale boundary value problem. The high performance of the homogenized model using TANN is illustrated through detailed comparisons with microstructural calculations at large scale. An excellent agreement is shown for a variety of monotonous and cyclic stress-strain paths.

Efthymios Papachristos, Ioannis Stefanou

Earthquakes are lethal and costly. This study aims at avoiding these catastrophic events by the application of injection policies retrieved through reinforcement learning. With the rapid growth of artificial intelligence, prediction-control problems are all the more tackled by function approximation models that learn how to control a specific task, even for systems with unmodeled/unknown dynamics and important uncertainties. Here, we show for the first time the possibility of controlling earthquake-like instabilities using state-of-the-art deep reinforcement learning techniques. The controller is trained using a reduced model of the physical system, i.e, the spring-slider model, which embodies the main dynamics of the physical problem for a given earthquake magnitude. Its robustness to unmodeled dynamics is explored through a parametric study. Our study is a first step towards minimizing seismicity in industrial projects (geothermal energy, hydrocarbons production, CO2 sequestration) while, in a second step for inspiring techniques for natural earthquakes control and prevention.

Filippo Masi, Ioannis Stefanou, Paolo Vannucci et al.

Machine Learning methods and, in particular, Artificial Neural Networks (ANNs) have demonstrated promising capabilities in material constitutive modeling. One of the main drawbacks of such approaches is the lack of a rigorous frame based on the laws of physics. This may render physically inconsistent the predictions of a trained network, which can be even dangerous for real applications. Here we propose a new class of data-driven, physics-based, neural networks for constitutive modeling of strain rate independent processes at the material point level, which we define as Thermodynamics-based Artificial Neural Networks (TANNs). The two basic principles of thermodynamics are encoded in the network's architecture by taking advantage of automatic differentiation to compute the numerical derivatives of a network with respect to its inputs. In this way, derivatives of the free-energy, the dissipation rate and their relation with the stress and internal state variables are hardwired in the network. Consequently, our network does not have to identify the underlying pattern of thermodynamic laws during training, reducing the need of large data-sets. Moreover the training is more efficient and robust, and the predictions more accurate. Finally and more important, the predictions remain thermodynamically consistent, even for unseen data. Based on these features, TANNs are a starting point for data-driven, physics-based constitutive modeling with neural networks. We demonstrate the wide applicability of TANNs for modeling elasto-plastic materials, with strain hardening and strain softening. Detailed comparisons show that the predictions of TANNs outperform those of standard ANNs. TANNs ' architecture is general, enabling applications to materials with different or more complex behavior, without any modification.