Meta-Auto-Decoder:a Meta-Learning-Based Reduced Order Model for Solving Parametric Partial Differential Equations

Many important problems in science and engineering require solving the so-called parametric partial differential equations(PDEs),i.e.,PDEs with different physical parameters,boundary conditions,shapes of computational domains,etc.Typical reduced order modeling techniques accelerate the solution of the parametric PDEs by projecting them onto a linear trial manifold constructed in the ofline stage.These methods often need a predefined mesh as well as a series of precomputed solution snapshots,and may struggle to balance between the efficiency and accuracy due to the limitation of the linear ansatz.Utilizing the nonlinear representation of neural networks(NNs),we propose the Meta-Auto-Decoder(MAD)to construct a nonlinear trial manifold,whose best possible performance is measured theoretically by the decoder width.Based on the meta-learning concept,the trial manifold can be learned in a mesh-free and unsupervised way during the pre-training stage.Fast adaptation to new(possibly heterogeneous)PDE parameters is enabled by searching on this trial manifold,and optionally fine-tuning the trial manifold at the same time.Extensive numerical experiments show that the MAD method exhibits a faster convergence speed without losing the accuracy than other deep learning-based methods.

Optimizing near-carbon-free nuclear energy systems:advances in reactor operation digital twin through hybrid machine learning algorithms for parameter identification and state estimation

Accurate and efficient online parameter identification and state estimation are crucial for leveraging digital twin simulations to optimize the operation of near-carbon-free nuclear energy systems.In previous studies,we developed a reactor operation digital twin(RODT).However,non-differentiabilities and discontinuities arise when employing machine learning-based surrogate forward models,challenging traditional gradient-based inverse methods and their variants.This study investigated deterministic and metaheuristic algorithms and developed hybrid algorithms to address these issues.An efficient modular RODT software framework that incorporates these methods into its post-evaluation module is presented for comprehensive comparison.The methods were rigorously assessed based on convergence profiles,stability with respect to noise,and computational performance.The numerical results show that the hybrid KNNLHS algorithm excels in real-time online applications,balancing accuracy and efficiency with a prediction error rate of only 1%and processing times of less than 0.1 s.Contrastingly,algorithms such as FSA,DE,and ADE,although slightly slower(approximately 1 s),demonstrated higher accuracy with a 0.3%relative L_2 error,which advances RODT methodologies to harness machine learning and system modeling for improved reactor monitoring,systematic diagnosis of off-normal events,and lifetime management strategies.The developed modular software and novel optimization methods presented offer pathways to realize the full potential of RODT for transforming energy engineering practices.

Adding-Point Strategy for Reduced-Order Hypersonic Aerothermodynamics Modeling Based on Fuzzy Clustering

Reduced order models（ROMs） based on the snapshots on the CFD high-fidelity simulations have been paid great attention recently due to their capability of capturing the features of the complex geometries and flow configurations. To improve the efficiency and precision of the ROMs, it is indispensable to add extra sampling points to the initial snapshots, since the number of sampling points to achieve an adequately accurate ROM is generally unknown in prior, but a large number of initial sampling points reduces the parsimony of the ROMs. A fuzzy-clustering-based adding-point strategy is proposed and the fuzzy clustering acts an indicator of the region in which the precision of ROMs is relatively low. The proposed method is applied to construct the ROMs for the benchmark mathematical examples and a numerical example of hypersonic aerothermodynamics prediction for a typical control surface. The proposed method can achieve a 34.5% improvement on the efficiency than the estimated mean squared error prediction algorithm and shows same-level prediction accuracy.

Sliding Mode Control Design via Reduced Order Model Approach

This paper presents a design of continuous-time sliding mode control for the higher order systems via reduced order model. It is shown that a continuous-time sliding mode control designed for the reduced order model gives similar performance for thc higher order system. The method is illustrated by numerical examples. The paper also introduces a technique for design of a sliding surface such that the system satisfies a cost-optimality condition when on the sliding surface.

Reduced order observer based identification of base isolated buildings

The objective of this study is to identify system parameters from the recorded response of base isolated buildings,such as USC hospital building,during the 1994 Northridge earthquake.Full state measurements are not available for identification.Additionally,the response is nonlinear due to the yielding of the lead-rubber bearings.Two new approaches are presented in this paper to solve the aforementioned problems.First,a reduced order observer is used to estimate the unmeasured states.Second,a least squares technique with time segments is developed to identify the piece-wise linear system properties.The observer is used to estimate the initial conditions needed for the time segmented identification.A series of equivalent linear system parameters are identified in different time segments.It is shown that the change in system parameters,such as frequencies and damping ratios,due to nonlinear behavior of the lead-rubber bearings,are reliably estimated using the presented technique.It is shown that the response was reduced due to yielding of the lead-rubber bearings and period lengthening.

DESIGNING REDUCED-ORDER CONTROLLERS OF MIXEDSENSITIVITY PROBLEM FOR FLIGHT CONTROL SYSTEMS

Based on linear matrix inequalities (LMI), the design method of reduced order controllers of mixed sensitivity problem is studied for flight control systems. It is shown that there exists a controller with order not greater than the difference between the generalized plant order and the number of independent control variables, if the mixed sensitivity problem is solvable for strict regular flight control plants. The proof is constructive, and an approach to design such a controller can be obtained in terms of a pair of feasible solution to the well known 3 LMI. Finally, an example of mixed sensitivity problem for a flight control system is given to demonstrate practice of the approach.

Enumeration Order Equivalence in Rational Numbers

The enumeration of elements of c.e. sets in the theory of computability and computational complexity has already been investigated. However, the order of this enumeration has received less attention. The enumeration orders of elements of c.e. sets by means of Turing machines on natural numbers are investigated. In this paper, we consider the enumeration orders of elements of c.e. sets on rational numbers. We present enumeration order reducibility and enumeration order equivalence on rational numbers and propose some lemmas and theorems on these concepts. Also, we show that the theories here hold for Rc and we could repeat the same theories in this domain, in a same way.

UNIFORM DIFFERENCE SCHEME FOR A SINGULARLY PERTURBED LINEAR 2ND ORDER HYPERBOLIC PROBLEM WITH ZEROTH ORDER REDUCED EQUATION

In this paper a singularly perturbed linear second order hyperbolic problem with zeroth order reduced equation is discussed. Firstly, an energy inequality of the solution and an estimate of the remainder term of the asymptotic solution are given. Then an exponentially fitted difference scheme is developed in an equidistant mesh. Finally, uniform convergence in small parameter is proved in the sense of discrete energy norm.

Assessment of a two-surface plasticity model for hexagonal materials

A computationally efficient two-surface plasticity model is assessed against crystal plasticity. Focus is laid on the mechanical behavior of magnesium alloys in the presence of ductility-limiting defects, such as voids. The two surfaces separately account for slip and twinning such that the constitutive formulation captures the evolving plastic anisotropy and evolving tension-compression asymmetry. For model identification, a procedure is proposed whereby the initial guess is based on a combination of experimental data and computationally intensive polycrystal calculations from the literature. In drawing direct comparisons with crystal plasticity, of which the proposed model constitutes a heuristically derived reduced-order model, the available crystal plasticity simulations are grouped in two datasets. A calibration set contains minimal data for both pristine and porous material subjected to one loading path. Then the two-surface model is assessed against a broader set of crystal plasticity simulations for voided unit cells under various stress states and two loading orientations. The assessment also includes microstructure evolution(rate of growth of porosity and void distortion). The ability of the two-surface model to capture essential features of crystal plasticity is analyzed along with an evaluation of computational cost. The prospects of using the model in guiding the development of physically sound damage models in Mg alloys are put forth in the context of high-throughput simulations.

A Numerical Study on Hydraulic Fracturing Problems via the Proper Generalized Decomposition Method

The hydraulic fracturing is a nonlinear,fluid-solid coupling and transient problem,in most cases it is always time-consuming to simulate this process numerically.In recent years,although many numerical methods were proposed to settle this problem,most of them still require a large amount of computer resources.Thus it is a high demand to develop more efficient numerical approaches to achieve the real-time monitoring of the fracture geometry during the hydraulic fracturing treatment.In this study,a reduced order modeling technique namely Proper Generalized Decomposition(PGD),is applied to accelerate the simulations of the transient,non-linear coupled system of hydraulic fracturing problem,to match this extremely tight response time constraint.The separability of the solution in space and time dimensions is studied for a simplified model problem.The solid and fluid equations are coupled explicitly by inverting the solid discrete problem,and a simple iterative procedure to handle the non-linear characteristic of the hydraulic fracturing problem is proposed in this work.Numeral validation illustrates that the results of PGD match well with these of standard finite element method in terms o f fracture opening and fluid pressure in the hydro-fracture.Moreover,after the off-line calculations,the numerical results can be obtained in real time.

Properties of Paths in ROBDD and Its Application to Signal Probability Calculations

The properties of the paths in an ROBDD representation of a Boolean function are presented and proved in the present paper, and the applications of ROBDD in calculating signal probability are also discussed. By this method, the troublesome calculation of the correlation among the nodes, which is caused by the re-convergent fan-out in digital system, can be avoided and power estimation can be faster than simulation-based method in [1].

Particle Swarm Optimization for the Design of H∞ Static Output Feedbacks

The design of H∞ reduced order controllers is known to be a non-convex optimization problem for which no generic solution exists. In this paper, the use of Particle Swarm Optimization （PSO） for the computation of H~ static output feedbacks is investigated. Two approaches are tested. In a first part, a probabilistic-type PSO algorithm is defined for the computation of discrete sets of stabilizing static output feedback controllers. This method relies on a technique for random sample generation in a given domain. It is therefore used for computing a suboptimal Ha static output feedback solution, In a second part, the initial optimization problem is solved by PSO, the decision variables being the feedback gains. Results are compared with standard reduced order problem solvers using the COMPIeib （Constraint Matrix-optimization Problem Library） benchmark examples and appear to be much than satisfactory, proving the great potential of PSO techniques.

Review of Proper Orthogonal Decomposition Method Applied to Computational Aeroelasticity

Over the recent years there has been an increased trend in the use of Reduced Order Models （ROM） for modeling the coupled aeroelastic system. Of all the ROM models, the Proper Orthogonal Decomposition Method （POD） has been the most widely used, reason being the relative simplicity of implementation and the physical insight that it offers towards the physical problem. In this paper we begin by briefly recalling the recent work using POD for the computational aeroelasticity followed by the mathematical formulation. Mathematical formulation is important as it provides understanding of how POD method works. Implementation issues related to the POD method are presented next. Since POD is an empirical technique therefore, it is marred by the robustness issues as is the case with all the ROMs. In the end the variations of POD method, developed over the years are presented along with the most recent trend of using hybrid ROM.

Data driven reduced modeling for fluidized bed with immersed tubes based on PCA and Bi-LSTM neural networksAuthor links open overlay panel

The fast and accurate reduced-order modeling of fluidized beds is a challenging task in the field of fluid dynamics,owing to their high dimensionality and nonlinear dynamic behavior.In this study,a nonintrusive reduced order modeling method,the reduced order model based on principal component analysis and bidirectional long short-term memory networks(PBLSTM ROM),was developed to capture complex spatio-temporal dynamics of fluidized beds.By combining principal component analysis and Bidirectional long-short-term memory networks,the PBLSTM ROM effectively extracted dynamic evolution information without any prior knowledge of governing equations,enabling reduced-order modeling of unsteady flow fields.The PBLSTM ROM was validated using the solid volume fraction and gas velocity flow fields of a fluidized bed with immersed tubes,showing superior performance over both the PLSTM and PANN ROMs in accurately capturing temporal changes in the fluidization fields,especially in the region near immersed tubes where severe fluctuations appear.Moreover,the PBLSTM ROM improved the simulation speed by five orders of magnitude compared to traditional computational fluid dynamics simulations.These findings suggest that the PBLSTM ROM presents a promising approach for analyzing the complex fluid flows in engineering practice.

NeuroPNM:Model reduction of pore network models using neural networks

Reacting particle systems play an important role in many industrial applications,for example biomass drying or the manufacturing of pharmaceuticals.The numerical modeling and simulation of such systems is therefore of great importance for an efficient,reliable,and environmentally sustainable operation of the processes.The complex thermodynamical,chemical,and flow processes that take place in the particles are a particular challenge in a simulation.Furthermore,typically a large number of particles is involved,rendering an explicit treatment of individual ones impossible in a reactor-level simulation.One approach for overcoming this challenge is to compute effective,physical parameters from single-particle,high-resolution simulations.This can be combined with model reduction methods if the dynamical behaviour of particles must be captured.Pore network models with their unrivaled resolution have thereby been used successfully as high-resolution models,for instance to obtain the macroscopic diffusion coeffcient of drying.Both parameter identification and model reduction have recently gained new impetus by the dramatic progress made in machine learning in the last decade.We report results on the use of neural networks for parameter identification and model reduction based on three-dimensional pore network models(PNM).We believe that our results provide a powerful complement to existing methodologies for reactor-level simulations with many thermally-thick particles.

A reduced order aerothermodynamic modeling framework for hypersonic vehicles based on surrogate and POD

Aerothermoelasticity is one of the key technologies for hypersonic vehicles. Accurate and efficient computation of the aerothermodynamics is one of the primary challenges for hypersonic aerothermoelastic analysis. Aimed at solving the shortcomings of engineering calculation, compu- tation fluid dynamics （CFD） and experimental investigation, a reduced order modeling （ROM） framework for aerothermodynamics based on CFD predictions using an enhanced algorithm of fast maximin Latin hypercube design is developed. Both proper orthogonal decomposition （POD） and surrogate are considered and compared to construct ROMs. Two surrogate approaches named Kriging and optimized radial basis function （ORBF） are utilized to construct ROMs. Furthermore, an enhanced algorithm of fast maximin Latin hypercube design is proposed, which proves to be helpful to improve the precisions of ROMs. Test results for the three-dimensional aerothermody- namic over a hypersonic surface indicate that： the ROMs precision based on Kriging is better than that by ORBF, ROMs based on Kriging are marginally more accurate than ROMs based on POD- Kriging. In a word, the ROM framework for hypersonic aerothermodynamics has good precision and efficiency.

Reduced order model for unsteady aerodynamic performance of compressor cascade based on recursive RBF

Based on Recursive Radial Basis Function(RRBF)neural network,the Reduced Order Model(ROM)of compressor cascade was established to meet the urgent demand of highly efficient prediction of unsteady aerodynamics performance of turbomachinery.One novel ROM called ASA-RRBF model based on Adaptive Simulated Annealing(ASA)algorithm was developed to enhance the generalization ability of the unsteady ROM.The ROM was verified by predicting the unsteady aerodynamics performance of a highly-loaded compressor cascade.The results show that the RRBF model has higher accuracy in identification of the dimensionless total pressure and dimensionless static pressure of compressor cascade under nonlinear and unsteady conditions,and the model behaves higher stability and computational efficiency.However,for the strong nonlinear characteristics of aerodynamic parameters,the RRBF model presents lower accuracy.Additionally,the RRBF model predicts with a large error in the identification of aerodynamic parameters under linear and unsteady conditions.For ASA-RRBF,by introducing a small-amplitude and highfrequency sinusoidal signal as validation sample,the width of the basis function of the RRBF model is optimized to improve the generalization ability of the ROM under linear unsteady conditions.Besides,this model improves the predicting accuracy of dimensionless static pressure which has strong nonlinear characteristics.The ASA-RRBF model has higher prediction accuracy than RRBF model without significantly increasing the total time consumption.This novel model can predict the linear hysteresis of dimensionless static pressure happened in the harmonic condition,but it cannot accurately predict the beat frequency of dimensionless total pressure.

Static aeroelastic analysis including geometric nonlinearities based on reduced order model

This paper describes a method proposed for modeling large deflection of aircraft in nonlinear aeroelastic analysis by developing reduced order model（ROM）.The method is applied for solving the static aeroelastic and static aeroelastic trim problems of flexible aircraft containing geometric nonlinearities;meanwhile,the non-planar effects of aerodynamics and follower force effect have been considered.ROMs are computational inexpensive mathematical representations compared to traditional nonlinear finite element method（FEM） especially in aeroelastic solutions.The approach for structure modeling presented here is on the basis of combined modal/finite element（MFE） method that characterizes the stiffness nonlinearities and we apply that structure modeling method as ROM to aeroelastic analysis.Moreover,the non-planar aerodynamic force is computed by the non-planar vortex lattice method（VLM）.Structure and aerodynamics can be coupled with the surface spline method.The results show that both of the static aeroelastic analysis and trim analysis of aircraft based on structure ROM can achieve a good agreement compared to analysis based on the FEM and experimental result.

Active Control Law Design for Flutter/LCO Suppression Based on Reduced Order Model Method

Active stability augmentation system is an attractive and promising technology to suppress flutter and limit cycle oscillation (LCO). In order to design a good active control law, the control plant model with low order and high accuracy must be provided, which is one of the most important key points. The traditional model is based on low fidelity aerodynamics model such as panel method, which is unsuitable for transonic flight regime. The physics-based high fidelity tools, reduced order model (ROM) and CFD/CSD coupled aeroservoelastic solver are used to design the active control law. The Volterra/ROM is applied to constructing the low order state space model for the nonlinear unsteady aerodynamics and static output feedback method is used to active control law design. The detail of the new method is demonstrated by the Goland+ wing/store system. The simulation results show that the effectiveness of the designed active augmentation system, which can suppress the flutter and LCO successfully.

Prediction on nonlinear mechanical performance of random particulate composites by a statistical second-order reduced multiscale approach

A novel statistical second-order reduced multiscale(SSRM)approach is established for nonlinear composite materials with random distribution of grains.For these composites considered in this work,the complex microstructure of grains,including their shape,orientation,size,spatial distribution,volume fraction and so on,results in changing of the macroscopic mechanical properties.The first-and second-order unit cell functions based on two-scale asymptotic expressions are constructed at first.Then,the expected homogenized parameters are defined,and the nonlinear homogenization equation on global structure is established,successively.Further,an effective reduced model format for analyzing second-order nonlinear unit cell problem with less computation cost is introduced in detail.Finally,some numerical examples for the materials with varying distribution models are evaluated and compared with the data by theoretical models and experimental results.These examples illustrate that the proposed SSRM approaches are effective for predicting the macroscopic properties of the random composite materials and supply a potential application in actual engineering computation.