Eduardo Camponogara

Jean Panaioti Jordanou, Eric Aislan Antonelo, Eduardo Camponogara et al.

Echo State Networks (ESN) are a type of Recurrent Neural Network that yields promising results in representing time series and nonlinear dynamic systems. Although they are equipped with a very efficient training procedure, Reservoir Computing strategies, such as the ESN, require high-order networks, i.e., many neurons, resulting in a large number of states that are magnitudes higher than the number of model inputs and outputs. A large number of states not only makes the time-step computation more costly but also may pose robustness issues, especially when applying ESNs to problems such as Model Predictive Control (MPC) and other optimal control problems. One way to circumvent this complexity issue is through Model Order Reduction strategies such as the Proper Orthogonal Decomposition (POD) and its variants (POD-DEIM), whereby we find an equivalent lower order representation to an already trained high dimension ESN. To this end, this work aims to investigate and analyze the performance of POD methods in Echo State Networks, evaluating their effectiveness through the Memory Capacity (MC) of the POD-reduced network compared to the original (full-order) ESN. We also perform experiments on two numerical case studies: a NARMA10 difference equation and an oil platform containing two wells and one riser. The results show that there is little loss of performance comparing the original ESN to a POD-reduced counterpart and that the performance of a POD-reduced ESN tends to be superior to a normal ESN of the same size. Also, the POD-reduced network achieves speedups of around $80\%$ compared to the original ESN.

João Zago, Eduardo Camponogara, Eric Antonelo

Neural networks achieved high performance over different tasks, i.e. image identification, voice recognition and other applications. Despite their success, these models are still vulnerable regarding small perturbations, which can be used to craft the so-called adversarial examples. Different approaches have been proposed to circumvent their vulnerability, including formal verification systems, which employ a variety of techniques, including reachability, optimization and search procedures, to verify that the model satisfies some property. In this paper we propose three novel reachability algorithms for verifying deep neural networks with ReLU activations. The first and third algorithms compute an over-approximation for the reachable set, whereas the second one computes the exact reachable set. Differently from previously proposed approaches, our algorithms take as input a V-polytope. Our experiments on the ACAS Xu problem show that the Exact Polytope Network Mapping (EPNM) reachability algorithm proposed in this work surpass the state-of-the-art results from the literature, specially in relation to other reachability methods.

Bruno Machado Pacheco, Laio Oriel Seman, Cezar Antonio Rigo et al.

This study investigates how to schedule nanosatellite tasks more efficiently using Graph Neural Networks (GNNs). In the Offline Nanosatellite Task Scheduling (ONTS) problem, the goal is to find the optimal schedule for tasks to be carried out in orbit while taking into account Quality-of-Service (QoS) considerations such as priority, minimum and maximum activation events, execution time-frames, periods, and execution windows, as well as constraints on the satellite's power resources and the complexity of energy harvesting and management. The ONTS problem has been approached using conventional mathematical formulations and exact methods, but their applicability to challenging cases of the problem is limited. This study examines the use of GNNs in this context, which has been effectively applied to optimization problems such as the traveling salesman, scheduling, and facility placement problems. More specifically, we investigate whether GNNs can learn the complex structure of the ONTS problem with respect to feasibility and optimality of candidate solutions. Furthermore, we evaluate using GNN-based heuristic solutions to provide better solutions (w.r.t. the objective value) to the ONTS problem and reduce the optimization cost. Our experiments show that GNNs are not only able to learn feasibility and optimality for instances of the ONTS problem, but they can generalize to harder instances than those seen during training. Furthermore, the GNN-based heuristics improved the expected objective value of the best solution found under the time limit in 45%, and reduced the expected time to find a feasible solution in 35%, when compared to the SCIP (Solving Constraint Integer Programs) solver in its off-the-shelf configuration

Eric Mochiutti, Eric Aislan Antonelo, Eduardo Camponogara

Echo State Networks (ESNs) are recurrent neural networks usually employed for modeling nonlinear dynamic systems with relatively ease of training. By incorporating physical laws into the training of ESNs, Physics-Informed ESNs (PI-ESNs) were proposed initially to model chaotic dynamic systems without external inputs. They require less data for training since Ordinary Differential Equations (ODEs) of the considered system help to regularize the ESN. In this work, the PI-ESN is extended with external inputs to model controllable nonlinear dynamic systems. Additionally, an existing self-adaptive balancing loss method is employed to balance the contributions of the residual regression term and the physics-informed loss term in the total loss function. The experiments with two nonlinear systems modeled by ODEs, the Van der Pol oscillator and the four-tank system, and with one differential-algebraic (DAE) system, an electric submersible pump, revealed that the proposed PI-ESN outperforms the conventional ESN, especially in scenarios with limited data availability, showing that PI-ESNs can regularize an ESN model with external inputs previously trained on just a few datapoints, reducing its overfitting and improving its generalization error (up to 92% relative reduction in the test error). Further experiments demonstrated that the proposed PI-ESN is robust to parametric uncertainties in the ODE equations and that model predictive control using PI-ESN outperforms the one using plain ESN, particularly when training data is scarce.

Jonas Ekeland Kittelsen, Eric Aislan Antonelo, Eduardo Camponogara et al.

Neural networks, while powerful, often lack interpretability. Physics-Informed Neural Networks (PINNs) address this limitation by incorporating physics laws into the loss function, making them applicable to solving Ordinary Differential Equations (ODEs) and Partial Differential Equations (PDEs). The recently introduced PINC framework extends PINNs to control applications, allowing for open-ended long-range prediction and control of dynamic systems. In this work, we enhance PINC for modeling highly nonlinear systems such as gas-lifted oil wells. By introducing skip connections in the PINC network and refining certain terms in the ODE, we achieve more accurate gradients during training, resulting in an effective modeling process for the oil well system. Our proposed improved PINC demonstrates superior performance, reducing the validation prediction error by an average of 67% in the oil well application and significantly enhancing gradient flow through the network layers, increasing its magnitude by four orders of magnitude compared to the original PINC. Furthermore, experiments showcase the efficacy of Model Predictive Control (MPC) in regulating the bottom-hole pressure of the oil well using the improved PINC model, even in the presence of noisy measurements.

Jean Panaioti Jordanou, Eduardo Camponogara, Eduardo Gildin

Linear Parameter Varying (LPV) Systems are a well-established class of nonlinear systems with a rich theory for stability analysis, control, and analytical response finding, among other aspects. Although there are works on data-driven identification of such systems, the literature is quite scarce in terms of works that tackle the identification of LPV models for large-scale systems. Since large-scale systems are ubiquitous in practice, this work develops a methodology for the local and global identification of large-scale LPV systems based on nonintrusive reduced-order modeling. The developed method is coined as DMD-LPV for being inspired in the Dynamic Mode Decomposition (DMD). To validate the proposed identification method, we identify a system described by a discretized linear diffusion equation, with the diffusion gain defined by a polynomial over a parameter. The experiments show that the proposed method can easily identify a reduced-order LPV model of a given large-scale system without the need to perform identification in the full-order dimension, and with almost no performance decay over performing a reduction, given that the model structure is well-established.

Bruno Machado Pacheco, Eduardo Camponogara

This paper introduces Physics-Informed Deep Equilibrium Models (PIDEQs) for solving initial value problems (IVPs) of ordinary differential equations (ODEs). Leveraging recent advancements in deep equilibrium models (DEQs) and physics-informed neural networks (PINNs), PIDEQs combine the implicit output representation of DEQs with physics-informed training techniques. We validate PIDEQs using the Van der Pol oscillator as a benchmark problem, demonstrating their efficiency and effectiveness in solving IVPs. Our analysis includes key hyperparameter considerations for optimizing PIDEQ performance. By bridging deep learning and physics-based modeling, this work advances computational techniques for solving IVPs, with implications for scientific computing and engineering applications.

Bruno Machado Pacheco, Laio Oriel Seman, Eduardo Camponogara

Maximizing oil production from gas-lifted oil wells entails solving Mixed-Integer Linear Programs (MILPs). As the parameters of the wells, such as the basic-sediment-to-water ratio and the gas-oil ratio, are updated, the problems must be repeatedly solved. Instead of relying on costly exact methods or the accuracy of general approximate methods, in this paper, we propose a tailor-made heuristic solution based on deep learning models trained to provide values to all integer variables given varying well parameters, early-fixing the integer variables and, thus, reducing the original problem to a linear program (LP). We propose two approaches for developing the learning-based heuristic: a supervised learning approach, which requires the optimal integer values for several instances of the original problem in the training set, and a weakly-supervised learning approach, which requires only solutions for the early-fixed linear problems with random assignments for the integer variables. Our results show a runtime reduction of 71.11% Furthermore, the weakly-supervised learning model provided significant values for early fixing, despite never seeing the optimal values during training.

Eric Aislan Antonelo, Eduardo Camponogara, Laio Oriel Seman et al.

Physics-informed neural networks (PINNs) impose known physical laws into the learning of deep neural networks, making sure they respect the physics of the process while decreasing the demand of labeled data. For systems represented by Ordinary Differential Equations (ODEs), the conventional PINN has a continuous time input variable and outputs the solution of the corresponding ODE. In their original form, PINNs do not allow control inputs, neither can they simulate for variable long-range intervals without serious degradation in their predictions. In this context, this work presents a new framework called Physics-Informed Neural Nets for Control (PINC), which proposes a novel PINN-based architecture that is amenable to control problems and able to simulate for longer-range time horizons that are not fixed beforehand, making it a very flexible framework when compared to traditional PINNs. Furthermore, this long-range time simulation of differential equations is faster than numerical methods since it relies only on signal propagation through the network, making it less computationally costly and, thus, a better alternative for simulation of models in Model Predictive Control. We showcase our proposal in the control of two nonlinear dynamic systems: the Van der Pol oscillator and the four-tank system.