In cyber-physical systems governed by nonlinear partial differential equations (PDEs), real-time control is often limited by sparse sensor data and high-dimensional system dynamics. Deep reinforcement learning (DRL) has shown promise for controlling such systems, but training DRL agents directly on full-order simulations is computationally intensive. This paper presents a sensor-driven, non-intrusive reduced-order modeling (NIROM) framework called FAE-CAE-LSTM, which combines convolutional and fully connected autoencoders with a long short-term memory (LSTM) network. The model compresses high-dimensional states into a latent space and captures their temporal evolution. A DRL agent is trained entirely in this reduced space, interacting with the surrogate built from sensor-like spatiotemporal measurements, such as pressure and velocity fields. A CNN-MLP reward estimator provides data-driven feedback without requiring access to governing equations. The method is tested on benchmark systems including Burgers’ equation, the Kuramoto–Sivashinsky equation, and flow past a circular cylinder; accuracy is further validated on flow past a square cylinder. Experimental results show that the proposed approach achieves accurate reconstruction, robust control, and significant computational speedup over traditional simulation-based training. These findings confirm the effectiveness of the FAE-CAE-LSTM surrogate in enabling real-time, sensor-informed, scalable DRL-based control of nonlinear dynamical systems.

Computation time and data storage is a significant challenge in every calculation process. Increasing computational speed by upgrading hardware and the introduction of new software are some of the techniques to overcome this challenge. One of the most interesting methods for fast computations is the reduced order frameworks. In this study, the aerodynamic coefficients of the NACA0012 airfoil in subsonic and supersonic flows have been reconstructed and estimated by a cost-efficient form of combined proper orthogonal decomposition–high-order singular value decomposition (POD-HOSVD) scheme. The initial data ensemble contains some members related to the variations of the angle of attack and Mach number. To reduce the computation time, the structure of the standard combined POD-HOSVD approach has been changed to a cost-efficient format. The present method is a grid independent formulation of standard combined POD-HOSVD for the fields with a large number of elements and several effective variables. Results indicate more than 90 percent reduction in the calculation time compares with computational fluid dynamics and standard combined POD-HOSVD methods for a subsonic flow field.

In this article, an improved reduced order modelling approach, based on the proper orthogonal decomposition (POD) method, is presented. After projecting the governing equations of flow dynamics along the POD modes, a dynamical system was obtained. Normally, the classical reduced order models do not predict accurate time variations of flow variables due to some reasons. The response of the dynamical system was improved using a calibration method based on a least-square optimization process. The calibration polynomial can be assumed as the pressure correction term which is vanished in projecting the Navier-Stokes equations along the POD modes. The above least- square procedure is a combination of POD method and the solution of an optimization problem. The obtained model can predict accurate time variations of flow field with high speed. For long time periods, the calibration term can be computed using a combined form of POD and Fourier modes. This extension is a totally new extension to this procedure which has recently been proposed by the authors. The results obtained from the calibrated reduced order model show close agreements to the benchmark DNS data, proving high accuracy of our model.

In this paper, a reduced order model is reconstructed for boundary control and excitation of the unsteady viscous Burgers equation. First, the standard reduced order proper orthogonal decomposition model, which has been extracted from the governing equations without control inputs, was evaluated and illustrated the satisfactory results in short time period. Two approaches are used to imply the effects of boundaries excitations and the related control routines. In the first, a source term was added to the governing equation of the reduced order dynamical system and was contributed as an expansion of the proper orthogonal decomposition modes without control input. For removing the inhomogeneities on the boundaries, the boundaries values are subtracted from all of the snapshots with an appropriate control input. The other approach is based on the rewriting of the diffusion term as an expanded form which contains the effect of boundaries values explicitly. In both approaches, the obtained reduced order models will contain two parts, the effect of system states and the influence of boundaries control functions. The results obtained from the reduced order model without the control inputs demonstrate a good agreement to the benchmark direct numerical simulations data and prove the high accuracy of the model.

The significant computational costs and time associated with accurate simulation of physical phenomena make the simulation of nonlinear systems based on differential equations impractical for real-time prediction. Deep learning with its high potential in understanding nonlinear and unknown phenomena can be a suitable alternative to equation-based modeling. However, the success of deep learning highly relies on the availability of large-scale labeled data. To solve this problem, weakly supervised learning helps us. This algorithm can train models using only a limited amount of labeled data. In this work, a new definition of the loss function was presented, which can greatly reduce our need to prepare labels for network training through weak supervision. We used this approach for 2D heat transfer modeling. The present work consists of two steps: (1) Extracting the equilibrium temperature pattern directly from only 400 thermal data and encoding it in a convolutional kernel that forms the loss function; and (2) unsupervised training of the model using this loss function instead of the labels without observing any thermal data. The effectiveness of the proposed model in terms of accuracy, the number of labeled data used, and the time required for training the network was evaluated and compared with three supervised models trained on large data sets. Despite using less data, our model achieved higher accuracy compared to a supervised model trained from direct observation of 5000 labeled thermal data (0.68% vs. 1.5% error), which has a longer training time than our model (20 vs. 12 hours); and the cGAN-based model despite using more than 10 times more labeled thermal data (0.68% vs. 1% error).