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1. Fast Accurate Full-Chip Dynamic Thermal Simulation with Fine Resolution Enabled by a Learning Method

   https://doi.org/10.1109/TCAD.2022.3229598  

   Jiang, Lin; Liu, Yu; Cheng, Ming-C (August 2023, IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems)

   The need for full-chip dynamic thermal simulation for effective runtime thermal management of multicore processors has been growing in recent years due to the rising demand for high-performance computing. In addition to simulation efficiency and accuracy, a high resolution is desirable in order to accurately predict crucial hot spots in the chip. This work investigates a simulation technique derived from proper orthogonal decomposition (POD) for full-chip dynamic thermal simulation of a multicore processor. The POD projects a heat transfer problem onto a mathematical space constituted by a finite set of basis functions (or POD modes) that are generated (or trained) by thermal solution data collected from direct numerical simulation (DNS). Accuracy and efficiency of the POD simulation technique influenced by the quality of thermal data are examined thoroughly, especially in the areas with high thermal gradients. The results show that if the POD modes are trained by good-quality data, the POD simulation offers an accurate prediction of the dynamic thermal distribution in the multicore processor with an extremely small degree of freedom (DoF). A reduction in computational time over four orders of magnitude, compared to the DNS, can be achieved for full-chip dynamic thermal simulation with a resolution as fine as the DNS. The study has also demonstrated that the POD approach can be used to rigorously verify the accuracy of solutions offered by DNS tools. A practical approach is proposed to further enhance the accuracy and efficiency of the proposed full-chip thermal simulation technique. 

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2. Rapid and Rigorous Learning Simulation Methodology with High Accuracy for Electronic and Optical Superlattices

   Veresko, Martin; Liu, Yu; Hou, Daqing; Cheng, Ming-Cheng (July 2024, 24th Int'l Conf. Computational Science (ICCS2024), Malaga, Spain)

   A rigorous physics-informed learning methodology is proposed for predictions of wave solutions and band structures in electronic and optical superlattice structures. The methodology is enabled by proper orthogonal decomposition (POD) and Galerkin projection of the wave equation. The approach solves the wave eigenvalue problem in POD space constituted by a finite set of basis functions (or POD modes). The POD ensures that the generated modes are optimized and tailored to the parametric variations of the system. Galerkin projection however enforces physical principles in the methodology to further enhance the accuracy and efficiency of the developed model. It has been demonstrated that the POD-Galerkin methodology offers an approach with a reduction in degrees of freedom by 4 orders of magnitude, compared to direct numerical simulation (DNS). A computing speedup near 15,000 times over DNS can be achieved with high accuracy for either of the superlattice structures if only the band structure is calculated without the wave solution. If both wave function solution and band structure are needed, a 2-order reduction in computational time can be achieved with a relative least square error (LSE) near 1%. When the training is incomplete or the desired eigenstates are slightly beyond the training bounds, an accurate prediction with an LSE near 1%-2% still can be reached if more POD modes are included. This reveals its remarkable learning ability to reach correct solutions with the guidance of physical principles provided by Galerkin projection. 

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3. Fast Simulations with High Accuracy for Periodic Quantum Nanostructures and Photonic Crystals Enabled by a Physics-Informed Projection Learning Methodology

   Veresko, Martin; Liu, Yu; Hou, Daqing; Cheng, Ming-Cheng (July 2024, XXXV IUPAP. Comp. Physics (CCP2024))

   Fast solution of wave propagation in periodic structures usually relies on simplified approaches, such as analytical methods, transmission line models, scattering matrix approaches, plane wave methods, etc. For complex multi-dimensional problems, computationally intensive direct numerical simulation (DNS) is always needed. This study demonstrates a fast and accurate simulation methodology enabled by a physics-based learning methodology, derived from proper orthogonal decomposition (POD) and Galerkin projection, for periodic quantum nanostructure and photonic crystals. POD is a projection-based method that generates optimal basis functions (or POD modes) via solution data collected from DNSs. This process trains the POD modes to adapt parametric variations of the system and offers the best least squares (LS) fit to the solution using the smallest number of modes. This is very different from other projection approaches, e.g., Fourier, Legendre, Bessel, Airy functions, etc., that adopt assumed basis functions selected for the problem based on the solution form. After generating the optimal POD modes, Galerkin projection of the wave equation onto each of the POD modes is performed to close the model and incorporate physical principles guided by the wave equation. Such a rigorous approach offers efficient simulations with high accuracy and exhibits the extrapolation ability in cases reasonably beyond the training bounds. The POD-Galerkin methodology is applied in this study to predict band structures and wave solutions for 2D periodic quantum-dot and photonic-lattice structures. The plane-wave approach is also included in a periodic quantum-dot structure to illustrate the superior performance of the POD-Galerkin methodology. The POD-Galerkin approach offers a 2-order computing speedup for both nanostructure and optical superlattices, compared to DNS, when solving both the wave solution and band structure. If the band structure is the only concern, a 4-order improvement in computational efficiency can be achieved. Fig. 1(a) shows the optical superlattice in a demonstration, where a unit cell includes 22 discs with diagonally symmetrical refractive indices and the background index n = 1. The POD modes for this case are trained by TE mode electric field data collected from DNSs with variation of diagonally symmetrical refractive indices. The LS error of the predicted electric field wave solution from the POD-Galerkin approach, shown in Fig. 1(b) compared to DNS, is below 1% with just 8 POD modes that offer a more than 4-order reduction in the degrees of freedom, compared to DNS. In addition, an extremely accurate prediction of band structure is illustrated in Fig. 1(c) with a maximum error below 0.1% in the entire Brillouin zone. 

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4. ERROR ESTIMATES FOR A POD METHOD FOR SOLVING VISCOUS G-EQUATIONS IN INCOMPRESSIBLE CELLULAR FLOWS

   https://doi.org/DOI. 10.1137/19M1241854  

   GU, HAOTIAN; XIN, JACK XIN; ZHANG, ZHIWEN (February 2021, SIAM journal on scientific computing)

   null (Ed.)

   The G-equation is a well-known model for studying front propagation in turbulent combustion. In this paper, we develop an efficient model reduction method for computing regular solutions of viscous G-equations in incompressible steady and time-periodic cellular flows. Our method is based on the Galerkin proper orthogonal decomposition (POD) method. To facilitate the algorithm design and convergence analysis, we decompose the solution of the viscous G-equation into a mean-free part and a mean part, where their evolution equations can be derived accordingly. We construct the POD basis from the solution snapshots of the mean-free part. With the POD basis, we can efficiently solve the evolution equation for the mean-free part of the solution to the viscous G-equation. After we get the mean-free part of the solution, the mean of the solution can be recovered. We also provide rigorous convergence analysis for our method. Numerical results for viscous G-equations and curvature G-equations are presented to demonstrate the accuracy and efficiency of the proposed method. In addition, we study the turbulent flame speeds of the viscous G-equations in incompressible cellular flows. 

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5. Regression Trees on Grassmann Manifold for Adapting Reduced-Order Models

   https://doi.org/10.2514/1.J062180  

   Liu, Xiao; Liu, Xinchao (March 2023, AIAA Journal)

   Low-dimensional and computationally less-expensive reduced-order models (ROMs) have been widely used to capture the dominant behaviors of high-4dimensional systems. An ROM can be obtained, using the well-known proper orthogonal decomposition (POD), by projecting the full-order model to a subspace spanned by modal basis modes that are learned from experimental, simulated, or observational data, i.e., training data. However, the optimal basis can change with the parameter settings. When an ROM, constructed using the POD basis obtained from training data, is applied to new parameter settings, the model often lacks robustness against the change of parameters in design, control, and other real-time operation problems. This paper proposes to use regression trees on Grassmann manifold to learn the mapping between parameters and POD bases that span the low-dimensional subspaces onto which full-order models are projected. Motivated by the observation that a subspace spanned by a POD basis can be viewed as a point in the Grassmann manifold, we propose to grow a tree by repeatedly splitting the tree node to maximize the Riemannian distance between the two subspaces spanned by the predicted POD bases on the left and right daughter nodes. Five numerical examples are presented to comprehensively demonstrate the performance of the proposed method, and compare the proposed tree-based method to the existing interpolation method for POD basis and the use of global POD basis. The results show that the proposed tree-based method is capable of establishing the mapping between parameters and POD bases, and thus adapt ROMs for new parameters. 

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