Control of Partial Differential Equations via Physics-Informed Neural Networks

Garcı́a-Cervera, Carlos J.; Kessler, Mathieu; Periago, Francisco

doi:10.1007/s10957-022-02100-4

Cited by 7 publications

(4 citation statements)

References 31 publications

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“…In our case, the solution 􏽢 u(x, t; P), which corresponds to the output of the NN, is constructed as described in [72], mainly:…”

Section: 3mentioning

confidence: 99%

Exploring Physics‐Informed Neural Networks for the Generalized Nonlinear Sine‐Gordon Equation

Deresse,

Dufera

2024

Applied Computational Intelligence and Soft Computing

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The nonlinear sine-Gordon equation is a prevalent feature in numerous scientific and engineering problems. In this paper, we propose a machine learning-based approach, physics-informed neural networks (PINNs), to investigate and explore the solution of the generalized non-linear sine-Gordon equation, encompassing Dirichlet and Neumann boundary conditions. To incorporate physical information for the sine-Gordon equation, a multiobjective loss function has been defined consisting of the residual of governing partial differential equation (PDE), initial conditions, and various boundary conditions. Using multiple densely connected independent artificial neural networks (ANNs), called feedforward deep neural networks designed to handle partial differential equations, PINNs have been trained through automatic differentiation to minimize a loss function that incorporates the given PDE that governs the physical laws of phenomena. To illustrate the effectiveness, validity, and practical implications of our proposed approach, two computational examples from the nonlinear sine-Gordon are presented. We have developed a PINN algorithm and implemented it using Python software. Various experiments were conducted to determine an optimal neural architecture. The network training was employed by using the current state-of-the-art optimization methods in machine learning known as Adam and L-BFGS-B minimization techniques. Additionally, the solutions from the proposed method are compared with the established analytical solutions found in the literature. The findings show that the proposed method is a computational machine learning approach that is accurate and efficient for solving nonlinear sine-Gordon equations with a variety of boundary conditions as well as any complex nonlinear physical problems across multiple disciplines.

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“…In our case, the solution 􏽢 u(x, t; P), which corresponds to the output of the NN, is constructed as described in [72], mainly:…”

Section: 3mentioning

confidence: 99%

Exploring Physics‐Informed Neural Networks for the Generalized Nonlinear Sine‐Gordon Equation

Deresse,

Dufera

2024

Applied Computational Intelligence and Soft Computing

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“…To avoid this issue, many studies considered the utilization of even small amounts of labeled data, 62,63 or targeted plants with boundary constraints for which the necessity of meta-learning is not significant. 64 In studies on the application of meta-learning techniques to control systems, [65][66][67][68] a main objective was quick and fine adaptation to various environmental conditions with small training data, but none of them was totally free from labeled data generation. As is well-known, transfer learning can also provide computational speed-up by initializing subnetworks.…”

Section: Introductionmentioning

confidence: 99%

“…However, using a PINN alone without collaborating with meta‐learning can cause significant degradation in predictions, leading to poor tracking performance in control. To avoid this issue, many studies considered the utilization of even small amounts of labeled data, 62,63 or targeted plants with boundary constraints for which the necessity of meta‐learning is not significant 64 . In studies on the application of meta‐learning techniques to control systems, 65–68 a main objective was quick and fine adaptation to various environmental conditions with small training data, but none of them was totally free from labeled data generation.…”

Section: Introductionmentioning

confidence: 99%

Event‐triggered model reference adaptive control system design for SISO plants using meta‐learning‐based physics‐informed neural networks without labeled data and transfer learning

Duanyai,

Song,

Konghuayrob

et al. 2024

Adaptive Control & Signal

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SummaryThis paper examines the controllability of a novel Lyapunov‐based model reference adaptive control (MRAC) system designed with a meta‐learning‐based physics‐informed neural network (MLPINN) for linear and nonlinear single‐input and single‐output (SISO) plants without labeled data (MLPINN‐MRAC system). It is devised with the benefits of several techniques: the integration of the identification process in a training mode into an online control mode (straightforward design); no labeled data generation for online identification by a physics‐informed neural network; the prevention of degradation in tracking performance by meta‐learning, as the system triggers a meta‐learning process only when an error threshold detects the deterioration (high efficiency and the reduction of computation cost); quick adaptation to new inputs and an updated control input for each sub‐time span by transfer learning. It is worth noting that the frequency of the meta‐learning event detection significantly affects the step response stability of the nonlinear plant. To achieve a better quality of the stabilization, more frequent event detection is necessary for both beginning and end intervals in control. Sixteen triggering events are enough to shape the acceptable step response of the nonlinear plant, and 44 triggering events achieve the plant's desired step response with its minor modeling error. While, as for the linear plant, a single triggering event is sufficient to attain its tolerable step response and modeling error, implying that intensive event detection is not critical in the identification. It is obvious that the MLPINN‐MRAC system functions well and is more beneficial and efficient for the nonlinear plant.

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“…Many researchers are interested in using neural networks to replace traditional numerical methods. A methodology and a set of guidelines for solving optimal control problems with PINNs are proposed in [22,23]. Barry-Straume et al use a two-stage framework to solve PDE-constrained optimization problems [24].…”

Section: Introductionmentioning

confidence: 99%

Deep Multi-Input and Multi-Output Operator Networks Method for Optimal Control of Pdes

Yong

Luo

Sun

2023

Preprint

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Control of Partial Differential Equations via Physics-Informed Neural Networks

Cited by 7 publications

References 31 publications

Exploring Physics‐Informed Neural Networks for the Generalized Nonlinear Sine‐Gordon Equation

Exploring Physics‐Informed Neural Networks for the Generalized Nonlinear Sine‐Gordon Equation

Event‐triggered model reference adaptive control system design for SISO plants using meta‐learning‐based physics‐informed neural networks without labeled data and transfer learning

Deep Multi-Input and Multi-Output Operator Networks Method for Optimal Control of Pdes

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