A numerical approach to the optimal control of thermally convective flows

The optimal control of thermally convective flows is usually modeled by an optimization problem with constraints of Boussinesq equations that consist of the Navier-Stokes equation and an advection-diffusion equation. This optimal control problem is challenging from both theoretical analysis and algorithmic design perspectives. For example, the nonlinearity and coupling of fluid flows and energy transports prevent direct applications of gradient type algorithms in practice. In this paper, we propose an efficient numerical method to solve this problem based on the operator splitting and optimization techniques. In particular, we employ the Marchuk-Yanenko method leveraged by the $L^2-$projection for the time discretization of the Boussinesq equations so that the Boussinesq equations are decomposed into some easier linear equations without any difficulty in deriving the corresponding adjoint system. Consequently, at each iteration, four easy linear advection-diffusion equations and two degenerated Stokes equations at each time step are needed to be solved for computing a gradient. Then, we apply the Bercovier-Pironneau finite element method for space discretization, and design a BFGS type algorithm for solving the fully discretized optimal control problem. We look into the structure of the problem, and design a meticulous strategy to seek step sizes for the BFGS efficiently. Efficiency of the numerical approach is promisingly validated by the results of some preliminary numerical experiments

The ADMM-PINNs Algorithmic Framework for Nonsmooth PDE-Constrained Optimization: A Deep Learning Approach

We study the combination of the alternating direction method of multipliers (ADMM) with physics-informed neural networks (PINNs) for a general class of nonsmooth partial differential equation (PDE)-constrained optimization problems, where additional regularization can be employed for constraints on the control or design variables. The resulting ADMM-PINNs algorithmic framework substantially enlarges the applicable range of PINNs to nonsmooth cases of PDE-constrained optimization problems. The application of the ADMM makes it possible to untie the PDE constraints and the nonsmooth regularization terms for iterations. Accordingly, at each iteration, one of the resulting subproblems is a smooth PDE-constrained optimization which can be efficiently solved by PINNs, and the other is a simple nonsmooth optimization problem which usually has a closed-form solution or can be efficiently solved by various standard optimization algorithms or pre-trained neural networks. The ADMM-PINNs algorithmic framework does not require to solve PDEs repeatedly, and it is mesh-free, easy to implement, and scalable to different PDE settings. We validate the efficiency of the ADMM-PINNs algorithmic framework by different prototype applications, including inverse potential problems, source identification in elliptic equations, control constrained optimal control of the Burgers equation, and sparse optimal control of parabolic equations

Accelerated primal-dual methods with enlarged step sizes and operator learning for nonsmooth optimal control problems

We consider a general class of nonsmooth optimal control problems with partial differential equation (PDE) constraints, which are very challenging due to its nonsmooth objective functionals and the resulting high-dimensional and ill-conditioned systems after discretization. We focus on the application of a primal-dual method, with which different types of variables can be treated individually and thus its main computation at each iteration only requires solving two PDEs. Our target is to accelerate the primal-dual method with either larger step sizes or operator learning techniques. For the accelerated primal-dual method with larger step sizes, its convergence can be still proved rigorously while it numerically accelerates the original primal-dual method in a simple and universal way. For the operator learning acceleration, we construct deep neural network surrogate models for the involved PDEs. Once a neural operator is learned, solving a PDE requires only a forward pass of the neural network, and the computational cost is thus substantially reduced. The accelerated primal-dual method with operator learning is mesh-free, numerically efficient, and scalable to different types of PDEs. The acceleration effectiveness of these two techniques is promisingly validated by some preliminary numerical results

Approximate and Weighted Data Reconstruction Attack in Federated Learning

Federated Learning (FL) is a distributed learning paradigm that enables multiple clients to collaborate on building a machine learning model without sharing their private data. Although FL is considered privacy-preserved by design, recent data reconstruction attacks demonstrate that an attacker can recover clients' training data based on the parameters shared in FL. However, most existing methods fail to attack the most widely used horizontal Federated Averaging (FedAvg) scenario, where clients share model parameters after multiple local training steps. To tackle this issue, we propose an interpolation-based approximation method, which makes attacking FedAvg scenarios feasible by generating the intermediate model updates of the clients' local training processes. Then, we design a layer-wise weighted loss function to improve the data quality of reconstruction. We assign different weights to model updates in different layers concerning the neural network structure, with the weights tuned by Bayesian optimization. Finally, experimental results validate the superiority of our proposed approximate and weighted attack (AWA) method over the other state-of-the-art methods, as demonstrated by the substantial improvement in different evaluation metrics for image data reconstructions

The Hard-Constraint PINNs for Interface Optimal Control Problems

We show that the physics-informed neural networks (PINNs), in combination with some recently developed discontinuity capturing neural networks, can be applied to solve optimal control problems subject to partial differential equations (PDEs) with interfaces and some control constraints. The resulting algorithm is mesh-free and scalable to different PDEs, and it ensures the control constraints rigorously. Since the boundary and interface conditions, as well as the PDEs, are all treated as soft constraints by lumping them into a weighted loss function, it is necessary to learn them simultaneously and there is no guarantee that the boundary and interface conditions can be satisfied exactly. This immediately causes difficulties in tuning the weights in the corresponding loss function and training the neural networks. To tackle these difficulties and guarantee the numerical accuracy, we propose to impose the boundary and interface conditions as hard constraints in PINNs by developing a novel neural network architecture. The resulting hard-constraint PINNs approach guarantees that both the boundary and interface conditions can be satisfied exactly and they are decoupled from the learning of the PDEs. Its efficiency is promisingly validated by some elliptic and parabolic interface optimal control problems

A two-stage numerical approach for the sparse initial source identification of a diffusion-advection equation

We consider the problem of identifying a sparse initial source condition to achieve a given state distribution of a diffusion-advection partial differential equation after a given final time. The initial condition is assumed to be a finite combination of Dirac measures. The locations and intensities of this initial condition are required to be identified. This problem is known to be exponentially ill-posed because of the strong diffusive and smoothing effects. We propose a two-stage numerical approach to treat this problem. At the first stage, to obtain a sparse initial condition with the desire of achieving the given state subject to a certain tolerance, we propose an optimal control problem involving sparsity-promoting and ill-posedness-avoiding terms in the cost functional, and introduce a generalized primal-dual algorithm for this optimal control problem. At the second stage, the initial condition obtained from the optimal control problem is further enhanced by identifying its locations and intensities in its representation of the combination of Dirac measures. This two-stage numerical approach is shown to be easily implementable and its efficiency in short time horizons is promisingly validated by the results of numerical experiments. Some discussions on long time horizons are also included

Deciphering the Key Pharmacological Pathways and Targets of Yisui Qinghuang Powder That Acts on Myelodysplastic Syndromes Using a Network Pharmacology-Based Strategy

Background. Yisui Qinghuang powder (YSQHP) is an effective traditional Chinese medicinal formulation used for the treatment of myelodysplastic syndromes (MDS). However, its pharmacological mechanism of action is unclear. Materials and Methods. In this study, the active compounds of YSQHP were screened using the traditional Chinese medicine systems pharmacology (TCMSP) and HerDing databases, and the putative target genes of YSQHP were predicted using the STITCH and DrugBank databases. Then, we further screened the correlative biotargets of YSQHP and MDS. Finally, the compound-target-disease (C-T-D) network was conducted using Cytoscape, while GO and KEGG analyses were conducted using R software. Furthermore, DDI-CPI, a web molecular docking analysis tool, was used to verify potential targets and pathways. Finally, binding site analysis was performed to identify core targets using MOE software. Results. Our results identified 19 active compounds and 273 putative target genes of YSQHP. The findings of the C-T-D network revealed that Rb1, CASP3, BCL2, and MAPK3 showed the most number of interactions, whereas indirubin, tryptanthrin, G-Rg1, G-Rb1, and G-Rh2 showed the most number of potential targets. The GO analysis showed that 17 proteins were related with STPK activity, PUP ligase binding, and kinase regulator activity. The KEGG analysis showed that PI3K/AKT, apoptosis, and the p53 pathways were the main pathways involved. DDI-CPI identified the top 25 proteins related with PI3K/AKT, apoptosis, and the p53 pathways. CASP8, GSK3B, PRKCA, and VEGFR2 were identified as the correlative biotargets of DDI-CPI and PPI, and their binding sites were found to be indirubin, G-Rh2, and G-Rf. Conclusion. Taken together, our results revealed that YSQHP likely exerts its antitumor effects by binding to CASP8, GSK3B, PRKCA, and VEGFR2 and by regulating the apoptosis, p53, and PI3K/AKT pathways

A new multiscale numerical characterization of mechanical properties of graphene-reinforced polymer-matrix composites

In the present study, a three-dimensional multiscale simulation method for analyzing the mechanical properties of graphene-reinforced polymer-matrix composites is proposed. The macroscopic and the atomistic scales are combined in the proposed finite element modeling approach. The macroscopic homogeneous isotropic model of the matrix and the interface is included in the representative volume element (RVE) of the composites. In the nanoscale analysis, a space frame structure of graphene is selected, the carbon atoms are described as nodes, and the carbon-carbon (C-C) covalent bonds are represented with nanoscale beams. The effect of graphene volume fraction and different inclined angles on the mechanical properties of the composites is investigated under axial tension. The simulation results showed that with the increase in the graphene volume fraction, the Young's modulus and shear modulus of the graphene-reinforced composites were increased significantly. The stress transfer in the interface of the composites was also analyzed using this multiscale approach

A finite element method to investigate the elastic properties of pillared graphene sheet under different conditions

In this paper, an investigation was carried out to understand the mechanical elastic properties of a newly developed 3D nanocarbon structure material known as PGS (pillared graphene sheet). The effect of various parameters such as the pillared distance, the chirality, the volume fraction and the inclination angle of carbon nanotube on the elastic moduli was studied using the finite element method. The commercially available finite element software ABAQUS was utilized for the modelling and analysis of this new structure. Several different models were developed to study the effect of the various parameters mentioned earlier. Some interesting conclusions were deduced from the finite element investigation. It was found that the pillared distance, chirality and volume fraction affected the modulus significantly. The change in volume fraction impacted a bigger influence on the Young's modulus Ez and shear modulus Gxy. In some instances, Ez was increased by approximately 40 times. In addition, the Young's modulus Ez was also found to be affected by the change in the inclination angle and was shown to increase with increasing angle