Ayoub Farkane

Ayoub Farkane, Mounir Ghogho, Mustapha Oudani et al.

Fluid mechanics is a fundamental field in engineering and science. Solving the Navier-Stokes equation (NSE) is critical for understanding the behavior of fluids. However, the NSE is a complex partial differential equation that is difficult to solve, and classical numerical methods can be computationally expensive. In this paper, we present an innovative approach for solving the NSE using Physics Informed Neural Networks (PINN) and several novel techniques that improve their performance. The first model is based on an assumption that involves approximating the velocity component by employing the derivative of a stream function. This assumption serves to simplify the system and guarantees that the velocity adheres to the divergence-free equation. We also developed a second more flexible model that approximates the solution without any assumptions. The proposed models can effectively solve two-dimensional NSE. Moreover, we successfully applied the second model to solve the three-dimensional NSE. The results show that the models can efficiently and accurately solve the NSE in three dimensions. These approaches offer several advantages, including high trainability, flexibility, and efficiency.

Ayoub Farkane, David Lassounon

Bacterial cancer therapy exploits anaerobic bacteria's ability to target hypoxia tumor regions, yet the interactions among tumor growth, bacterial colonization, oxygen levels, immunosuppressive cytokines, and bacterial communication remain poorly quantified. We present a mathematical model of five coupled nonlinear reaction-diffusion equations in a two-dimensional tissue domain. We proved the global well-posedness of the model and identified its steady states to analyze stability. Furthermore, a physics-informed neural network (PINN) solves the system without a mesh and without requiring extensive data. It provides convergence guarantees by combining residual stability and Sobolev approximation error bounds. This results in an overall error rate of O(n^-2 ln^4(n) + N^-1/2), which depends on the network width n and the number of collocation points N. We conducted several numerical experiments, including predicting the tumor's response to therapy. We also performed a sensitivity analysis of certain parameters. The results suggest that long-term therapeutic efficacy may require the maintenance of hypoxia regions in the tumor, or using bacteria that tolerate oxygen better, may be necessary for long-lasting tumor control.

Ayoub Farkane, Mohamed Boutayeb, Mustapha Oudani et al.

Accurate knowledge of the state variables in a dynamical system is critical for effective control, diagnosis, and supervision, especially when direct measurements of all states are infeasible. This paper presents a novel approach to designing software sensors for nonlinear dynamical systems expressed in their most general form. Unlike traditional model-based observers that rely on explicit transformations or linearization, the proposed framework integrates neural networks with adaptive Sliding Mode Control (SMC) to design a robust state observer under a less restrictive set of conditions. The learning process is driven by available sensor measurements, which are used to correct the observer's state estimate. The training methodology leverages the system's governing equations as a physics-based constraint, enabling observer synthesis without access to ground-truth state trajectories. By employing a time-varying gain matrix dynamically adjusted by the neural network, the observer adapts in real-time to system changes, ensuring robustness against noise, external disturbances, and variations in system dynamics. Furthermore, we provide sufficient conditions to guarantee estimation error convergence, establishing a theoretical foundation for the observer's reliability. The methodology's effectiveness is validated through simulations on challenging examples, including systems with non-differentiable dynamics and varying observability conditions. These examples, which are often problematic for conventional techniques, serve to demonstrate the robustness and broad applicability of our approach. The results show rapid convergence and high accuracy, underscoring the method's potential for addressing complex state estimation challenges in real-world applications.

Ayoub Farkane, Mohamed Boutayeb, Mustapha Oudani et al.

State estimation for nonlinear dynamical systems is a critical challenge in control and engineering applications, particularly when only partial and noisy measurements are available. This paper introduces a novel Adaptive Physics-Informed Neural Network-based Observer (PINN-Obs) for accurate state estimation in nonlinear systems. Unlike traditional model-based observers, which require explicit system transformations or linearization, the proposed framework directly integrates system dynamics and sensor data into a physics-informed learning process. The observer adaptively learns an optimal gain matrix, ensuring convergence of the estimated states to the true system states. A rigorous theoretical analysis establishes formal convergence guarantees, demonstrating that the proposed approach achieves uniform error minimization under mild observability conditions. The effectiveness of PINN-Obs is validated through extensive numerical simulations on diverse nonlinear systems, including an induction motor model, a satellite motion system, and benchmark academic examples. Comparative experimental studies against existing observer designs highlight its superior accuracy, robustness, and adaptability.