Local Minima Research Articles

Review of physics-informed machine-learning inversion of geophysical data

We review five types of physics-informed machine-learning (PIML) algorithms for inversion and modeling of geophysical data. Such algorithms use the combination of a data-driven machine-learning (ML) method and the equations of physics to model or invert geophysical data (or both). By incorporating the constraints of physics, PIML algorithms can effectively reduce the size of the solution space for ML models, enabling them to be trained on smaller data sets. This is especially advantageous in scenarios in which data availability may be limited or expensive to obtain. In this review, we restrict the physics to be that from the governing wave equation, either as a constraint that must be satisfied or by using numerical solutions of the wave equation for modeling and inversion. This approach ensures that the resulting models adhere to physical principles while leveraging the power of ML to analyze and interpret complex geophysical data. There are several potential benefits of PIML compared to standard numerical modeling or inversion of seismic data computed by, for example, finite-difference solutions to the wave equation. 1) Empirical tests suggest that PIML algorithms constrained by the physics of wave propagation can sometimes resist getting stuck in a local minima compared with standard full-waveform inversion (FWI). 2) After the weights of the neural network are found by training, then the forward and inverse operations by PIML can be more than several orders of magnitude more efficient than FWI. However, the computational cost for general training can be enormous. 3) If the ML inversion operator [Formula: see text] is locally trained on a small portion of the recorded data [Formula: see text], then there is sometimes no need for millions of training examples that aim for global generalization of [Formula: see text]. The benefit is that the locally trained [Formula: see text] can be used to economically invert the remaining test data [Formula: see text] for the true velocity [Formula: see text], where [Formula: see text] can comprise more than 90% of the recorded data.

A piecewise smooth version of the Griewank function

The Griewank test function for global unconstrained optimization has multiple local minima clustered around the global minimum at the origin. A new version of this test function is proposed that has a similar structure, but whose behavior at the local minima and maxima is non-smooth. This piecewise smooth version of the Griewank function represents an abs-factorable test case of objective functions for global non-smooth optimization as, for example, observed in the training of neural networks.

Mixed H2S and H2O Clusters─New Insights into Dispersion-Dominated Hydrogen Bonding.

Here, we report the results of an IR spectroscopy study on heteroclusters of H2S and H2O and several of their isotopomers using mass-selective IR spectroscopy in superfluid helium nanodroplets in the range of 2560-2800 cm-1. Based on DFT calculations on the B3LYP-D3/6-311++G(d,p) level of theory, we were able to assign the experimentally observed O-D stretching bands to heterodimer and heterotrimer clusters. Since no bands of the S-H-bound conformer HSH···OH2 could be observed, we were able to determine the O-H-bound conformer HOH···SH2 to be the global minimum structure. A trapping of the local minima in helium nanodroplets was not observed. This is in line with the weaker hydrogen bond expected for H2S complexes. In these clusters, the interaction energy is expected to be more dominated by dispersion and less dictated by highly directional electrostatic forces.

Surface Cutting and Flattening to Target Shapes

AbstractWe introduce a novel framework for surface cutting and flattening, aiming to align the boundary of planar parameterization with a target shape. Diverging from traditional methods focused on minimizing distortion, we intend to also achieve shape similarity between the parameterized mesh and a specific planar target, which is important in some applications of art design and texture mapping. However, with existing methods commonly limited to ellipsoidal surfaces, it still remains a challenge to solve this problem on general surfaces. Our framework models the general case as a joint optimization of cuts and parameterization, guided by a novel metric assessing shape similarity. To circumvent the common issue of local minima, we introduce an extra global seam updating strategy which is guided by the target shape. Experimental results show that our framework not only aligns with previous approaches on ellipsoidal surfaces but also achieves satisfactory results on more complex ones.

FastFlow: GPU Acceleration of Flow and Depression Routing for Landscape Simulation

AbstractTerrain analysis plays an important role in computer graphics, hydrology and geomorphology. In particular, analyzing the path of material flow over a terrain with consideration of local depressions is a precursor to many further tasks in erosion, river formation, and plant ecosystem simulation. For example, fluvial erosion simulation used in terrain modeling computes water discharge to repeatedly locate erosion channels for soil removal and transport. Despite its significance, traditional methods face performance constraints, limiting their broader applicability.In this paper, we propose a novel GPU flow routing algorithm that computes the water discharge in 𝒪(log n) iterations for a terrain with n vertices (assuming n processors). We also provide a depression routing algorithm to route the water out of local minima formed by depressions in the terrain, which converges in 𝒪(log2 n) iterations. Our implementation of these algorithms leads to a 5× speedup for flow routing and 34 × to 52 × speedup for depression routing compared to previous work on a 10242 terrain, enabling interactive control of terrain simulation.

Mitigating local minima in extracting optimal parameters for photovoltaic models: An optimizer leveraging multiple initial populations (OLMIP)

The accurate identification of parameters in photovoltaic models is of paramount importance to accurately predict the electrical behavior of photovoltaic systems and improve their performance. Nowadays, this task poses a multimodal optimization problem, aiming to find the best combination of parameters that fit the model to real data. This article presents a proposal for an Optimizer Leveraging Multiple Initial Populations (OLMIP) that aims to achieve optimal solutions while effectively avoiding undesirable local optima. By utilizing a separate evolution strategy involving four distinct initial populations, followed by the construction of an elite population, the algorithm can explore multiple regions of the search space and escape local minima. Experiments were conducted with four models: the single diode, double diode, triple diode from RTC France cell, and Photowatt-PWP201 module. The mean squared errors obtained are 9.860219E-04, 9.824849E-04, 9.824849E-04, and 2.425075 E−03, respectively. These results indicate that the algorithm achieves superior or comparable accuracy to that of six competitors. Furthermore, the statistical analysis of the results, including the Wilcoxon and Friedman tests, confirms the robustness and effectiveness of the approach used for parameter estimation in photovoltaic systems. These findings open up new prospects for the improvement and optimization of photovoltaic systems.

Robust full waveform inversion with deep Hessian deblurring

SUMMARY Full waveform inversion (FWI) is a technique widely used in geophysics to obtain high-resolution subsurface velocity models from waveform seismic data. Due to its large computation cost, most flavours of FWI rely only on the computation of the gradient of the loss function to estimate the update direction, therefore ignoring the contribution of the Hessian. Depending on the level of computational resources one can afford, an approximate of the inverse of the Hessian can be calculated and used to speed up the convergence of FWI towards the global (or a plausible local) minimum. In this work, we propose to use an approximate Hessian computed from a linearization of the wave equation as commonly done in least-squares migration. More precisely, we rely on the link between a migrated image and a doubly migrated image (i.e. an image obtained by demigration–migration of the migrated image) to estimate the inverse of the Hessian. However, instead of using non-stationary compact filters to link the two images and approximate the Hessian, we propose to use a deep neural network to directly learn the mapping between the FWI gradient (output) and its Hessian (blurred) counterpart (input). By doing so, the network learns to act as an approximate inverse Hessian: as such, when the trained network is applied to the FWI gradient, an enhanced update direction is obtained, which is shown to be beneficial for the convergence of FWI. The weights of the trained (deblurring) network are then transferred to the next FWI iteration to expedite convergence. We demonstrate the effectiveness of the proposed approach on two synthetic data sets and a field data set.

Induction of orbital currents and Kapitza stabilization in superconducting circuits with Laguerre-Gaussian microwave beams

We investigate the effects of a Laguerre-Gaussian (LG) beam on the superconducting state. We show that the vortex angular momentum of a LG beam affects the superconducting state and induces currents. The induction of the current by microwave radiation is illustrated on a Josephson loop and SQUID devices. In particular, we establish that coupling a dc SQUID to the ac magnetic flux of a LG beam can stabilize the π phase in the SQUID. This can happen via developing a global or local minimum in the effective potential at π. In the latter case, this happens via the Kapitza mechanism. Published by the American Physical Society 2024

A circuit-generated quantum subspace algorithm for the variational quantum eigensolver.

Recent research has shown that wavefunction evolution in real and imaginary time can generate quantum subspaces with significant utility for obtaining accurate ground state energies. Inspired by these methods, we propose combining quantum subspace techniques with the variational quantum eigensolver (VQE). In our approach, the parameterized quantum circuit is divided into a series of smaller subcircuits. The sequential application of these subcircuits to an initial state generates a set of wavefunctions that we use as a quantum subspace to obtain high-accuracy groundstate energies. We call this technique the circuit subspace variational quantum eigensolver (CSVQE) algorithm. By benchmarking CSVQE on a range of quantum chemistry problems, we show that it can achieve significant error reduction in the best case compared to conventional VQE, particularly for poorly optimized circuits, greatly improving convergence rates. Furthermore, we demonstrate that when applied to circuits trapped at local minima, CSVQE can produce energies close to the global minimum of the energy landscape, making it a potentially powerful tool for diagnosing local minima.

An Innovative Enhanced JAYA Algorithm for the Optimization of Continuous and Discrete Problems

Metaheuristic algorithms have gained popularity in the past decade due to their remarkable ability to address various optimization challenges. Among these, the JAYA algorithm has emerged as a recent contender that demonstrates strong performance across different optimization problems, largely attributed to its simplicity. However, real-world problems have become increasingly complex in today’s era, creating a demand for more robust and effective solutions to tackle these intricate challenges and achieve outstanding results. This article proposes an enhanced JAYA (EJAYA) method that addresses its inherent shortcomings, resulting in improved convergence and search capabilities when dealing with diverse problems. The current study evaluates the performance of the proposed optimization methods on both continuous and discontinuous problems. Initially, EJAYA is applied to solve 20 prominent test functions and is validated by comparison with other contemporary algorithms in the literature, including moth–flame optimization, particle swarm optimization, the dragonfly algorithm, and the sine–cosine algorithm. The effectiveness of the proposed approach in discrete scenarios is tested using feature selection and compared to existing optimization strategies. Evaluations across various scenarios demonstrate that the proposed enhancements significantly improve the JAYA algorithm’s performance, facilitating escape from local minima, achieving faster convergence, and expanding the search capabilities.

Decoding LDPC Codes with a Regular Check Matrix Structure Using the Bit Flip Method

Problem statement: low-density LDPC codes have become very popular in modern communication systems, which make it possible to implement decoders for them in practice. In conditions of severe limitations of equipment performance, the task of searching and researching the characteristics of decoding algorithms for these codes is urgent. The aim of the work is to find a simplified algorithm for decoding LDPC codes and to study the noise immunity of the proposed algorithm. The methods used: to solve this problem computer simulation is used to compare the noise immunity of known decoding algorithms and the proposed algorithm. Scientific novelty: the noise immunity characteristics of the proposed Bit Flip algorithm for decoding LDPC code with a regular structure of the check matrix using gradient descent and the procedure for escaping the local minimum are investigated. Result: a specific variant of the Bit Flip algorithm for LDPC code with a regular structure check matrix using gradient descent and a procedure for escaping the local minimum is proposed. The results of computer simulation are presented. Theoretical significance: the characteristics of noise immunity for the proposed variant of the LDPC code decoding algorithm with a regular structure of the check matrix are investigated. Practical significance: when using the proposed LDPC code decoding algorithm in the receiver, it is possible to calculate the parameters of the communication system based on the available dependencies of the error coefficient on the signal-to-noise ratio.

Random Natural Gradient

Hybrid quantum-classical algorithms appear to be the most promising approach for near-term quantum applications. An important bottleneck is the classical optimization loop, where the multiple local minima and the emergence of barren plateaux make these approaches less appealing. To improve the optimization the Quantum Natural Gradient (QNG) method \cite{stokes2020quantum} was introduced – a method that uses information about the local geometry of the quantum state-space. While the QNG-based optimization is promising, in each step it requires more quantum resources, since to compute the QNG one requires O(m2) quantum state preparations, where m is the number of parameters in the parameterized circuit. In this work we propose two methods that reduce the resources/state preparations required for QNG, while keeping the advantages and performance of the QNG-based optimization. Specifically, we first introduce the Random Natural Gradient (RNG) that uses random measurements and the classical Fisher information matrix (as opposed to the quantum Fisher information used in QNG). The essential quantum resources reduce to linear O(m) and thus offer a quadratic "speed-up", while in our numerical simulations it matches QNG in terms of accuracy. We give some theoretical arguments for RNG and then benchmark the method with the QNG on both classical and quantum problems. Secondly, inspired by stochastic-coordinate methods, we propose a novel approximation to the QNG which we call Stochastic-Coordinate Quantum Natural Gradient that optimizes only a small (randomly sampled) fraction of the total parameters at each iteration. This method also performs equally well in our benchmarks, while it uses fewer resources than the QNG.

TOP2DFVT: An Efficient Matlab Implementation for Topology Optimization based on the Finite-Volume Theory.

The finite-volume theory has shown to be numerically efficient and stable for topology optimization of continuum elastic structures. The significant features of this numerical technique are the local satisfaction of equilibrium equations and the employment of compatibility conditions along edges in a surface-averaged sense. These are essential properties to adequately mitigate some numerical instabilities in the gradient version of topology optimization algorithms, such as checkerboard, mesh dependence, and local minima issues. Several computational tools have been proposed for topology optimization employing analysis domains discretized with essential features for finite-element approaches. However, this is the first contribution to offer a platform to generate optimized topologies by employing a Matlab code based on the finite-volume theory for compliance minimization problems. The Top2DFVT provides a platform to perform 2D topology optimization of structures in Matlab, from domain initialization for structured meshes to data post-processing. This contribution represents a significant advancement over earlier publications on topology optimization based on the finite-volume theory, which needed more efficient computational tools. Moreover, the Top2DFVT algorithm incorporates SIMP and RAMP material interpolation schemes alongside sensitivity and density filtering techniques, culminating in a notably enhanced optimization tool. The application of this algorithm to various illustrative cases confirms its efficacy and underscores its potential for advancing the field of structural optimization.

Dynamical signatures of discontinuous phase transitions: How phase coexistence determines exponential versus power-law scaling.

There are conflicting reports in the literature regarding the finite-size scaling of the Liouvillian gap and dynamical fluctuations at discontinuous phase transitions, with various studies reporting either exponential or power-law behavior. We clarify this issue by employing large deviation theory. We distinguish two distinct classes of discontinuous phase transitions that have different dynamical properties. The first class is associated with phase coexistence, i.e., the presence of multiple stable attractors of the system dynamics (e.g., local minima of the free-energy functional) in a finite phase diagram region around the phase transition point. In that case, one observes asymptotic exponential scaling related to stochastic switching between attractors (though the onset of exponential scaling may sometimes occur for very large system sizes). In the second class, there is no phase coexistence away from the phase transition point, while at the phase transition point itself there are infinitely many attractors. In that case, one observes power-law scaling related to the diffusive nature of the system relaxation to the stationary state.

Estimating reaction barriers with deep reinforcement learning1

Stable states in complex systems correspond to local minima on the associated potential energy surface. Transitions between these local minima govern the dynamics of such systems. Precisely determining the transition pathways in complex and high-dimensional systems is challenging because these transitions are rare events, and isolating the relevant species in experiments is difficult. Most of the time, the system remains near a local minimum, with rare, large fluctuations leading to transitions between minima. The probability of such transitions decreases exponentially with the height of the energy barrier, making the system’s dynamics highly sensitive to the calculated energy barriers. This work aims to formulate the problem of finding the minimum energy barrier between two stable states in the system’s state space as a cost-minimization problem. It is proposed to solve this problem using reinforcement learning algorithms. The exploratory nature of reinforcement learning agents enables efficient sampling and determination of the minimum energy barrier for transitions.

FWI-WF: Robust full-waveform inversion using hybrid Wasserstein-1 and Fourier metrics

The performance of full-waveform inversion (FWI) in constructing high-resolution subsurface models is closely related to the design of mismatch functions. The least-squares norm ([Formula: see text]) is commonly used; however, it is prone to local minima when high-quality initial guess and low-frequency data are unavailable. The Wasserstein-1 ([Formula: see text]) metric captures time shifts more effectively, but it may be plagued by imprecise deep structures. The Fourier metric leverages power spectra from simulated and observed data, offering higher-resolution updates near solutions. We develop a progressive waveform inversion method called FWI-WF using [Formula: see text] and Fourier metrics. Specifically, in the early stage of inversion, we apply greater weight to the [Formula: see text] metric for constructing a good background model and avoiding falling into local minima. Then, the Fourier metric gradually dominates to refine edges and deep structures, providing high-resolution inversion results. During the optimization process, we use automatic differentiation to improve inversion efficiency. Experimental results on three baseline geologic models indicate that FWI-WF outperforms three state-of-the-art methods.

Comparative fitting methodology of cure kinetics models based on differential scanning calorimetry

The optimization of composite cure cycles is critical in materials engineering, requiring a deep understanding of the material’s cure kinetics. This study employs Differential Scanning Calorimetry (DSC) coupled with advanced fitting algorithms to determine the most accurate cure kinetics models for optimizing time-temperature profiles. Given the wide variability in model selection and fitting approaches in existing literature, our research systematically compares several cure kinetics models and their fitting methodologies to identify the most effective strategies. We highlight the advantages and limitations of each approach, with a particular focus on the robustness of the Levenberg–Marquardt (LM) algorithm compared to the Trust-Region-Reflective (TRR) algorithm. Given the existence of multiple local minima in the objective function, a global approach (genetic algorithm) is employed. Our findings reveal that, although the TRR and LM algorithms are faster, they are less accurate than the genetic algorithm. Moreover, the Kamal-Sourour model shows the highest prediction accuracy among the tested models. This comparative analysis provides a systematic approach and crucial insights into selecting the optimal cure kinetic model and corresponding parameters, based on experimental data, allowing for enhancing the efficiency and accuracy of composite curing processes.

Distribution of Najadaceae Species in Trans-Baikal Area Due to Climatic Factors

According to data of the late 20th century in the Trans-Baikal and Baikal areas, the family Najadaceae is represented by four species known in few localities: Caulinia flexilis (4 localities), C. minor (3), and Najas major (3), N. marina (4). All of them are annuals, r-strategists, which are characterized by fluctuations in abundance. Since the 21st century, these species have been found in 20 more localities, mainly situated within the Baikal Depression. In 2015, when performing monitoring studies of aquatic flora, three more new localities of Caulinia flexilis, C. minor, and Najas major were found in the basin of the Amur River. Distribution maps of the Najadaceae species in the region were compiled, and the ecological characteristics of their habitats were given. In Trans-Baikal area, since the middle of the 20th century, warming has occurred due to an increase in the average annual temperature. For the last 70 years, aridization has increased in southern regions, which changes the habitat conditions in waterbodies. Findings of these Najadaceae species in Trans-Baikal area coincided with the periods of increase in mean annual air temperatures and with local minimums of precipitation. An increase in the number of findings of the Najadaceae species since the beginning of the 21st century, both in Trans-Baikal area and in other regions of Russia, has been occurring against the background of an increase in the average annual air temperature. In Trans-Baikal area, the trend of increasing of temperatures in the warm season since the beginning of the 21st century is favorable for these species and may contribute to their further distribution.

Application of improved Jellyfish search algorithm for 9-parameters cell extraction and GMPPT in PV systems

Optimization algorithms are essential tools used to address real-world problems through minimization or maximization processes. In the context of photovoltaic (PV) systems, various optimization algorithms have been employed to extract solar cell parameters using minimization techniques, and to identify the maximum power point (MPP) using maximization approaches. However, under partial shading conditions or during rapidly changing irradiance, many of these algorithms tend to get trapped in local minima, failing to locate the global maximum power point (GMPPT). This shortcoming leads to significant energy losses from PV cells. Similarly, the extraction of solar cell parameters is often compromised by the random search behavior and inadequate exploration and exploitation capabilities of many existing algorithms. The Jellyfish Search Optimization (JSO) algorithm is a recent, parameter-free method developed to tackle a wide range of optimization problems. Despite its innovative design, JSO is prone to premature convergence and exhibits a longer convergence time. To address these limitations, this paper proposes a modified version of the JSO algorithm, which integrates the state-of-the-art features of JSO with the Lévy flight characteristic from the cuckoo search algorithm. This combination aims to achieve a better balance between exploration and exploitation. To validate the effectiveness of the proposed Improved Jellyfish Search Optimization (IJSO) algorithm in PV systems, extensive experiments were conducted. These experiments included benchmarking with complex test functions, extracting cell parameters using various solar cell models, and maximizing energy output from PV systems under different environmental conditions. The results demonstrate that the proposed IJSO algorithm effectively enhances parameter extraction and global maximum power point tracking, making it a promising solution for optimizing energy extraction from PV cells in dynamic conditions.

Solving convex optimization problems via a second order dynamical system with implicit Hessian damping and Tikhonov regularization

AbstractThis paper deals with a second order dynamical system with a Tikhonov regularization term in connection to the minimization problem of a convex Fréchet differentiable function. The fact that beside the asymptotically vanishing damping we also consider an implicit Hessian driven damping in the dynamical system under study allows us, via straightforward explicit discretization, to obtain inertial algorithms of gradient type. We show that the value of the objective function in a generated trajectory converges rapidly to the global minimum of the objective function and depending the Tikhonov regularization parameter the generated trajectory converges weakly to a minimizer of the objective function or the generated trajectory converges strongly to the element of minimal norm from the $$\mathop {\text {argmin}}$$ argmin set of the objective function. We also obtain the fast convergence of the velocities towards zero and some integral estimates. Our analysis reveals that the Tikhonov regularization parameter and the damping parameters are strongly correlated, there is a setting of the parameters that separates the cases when weak convergence of the trajectories to a minimizer and strong convergence of the trajectories to the minimal norm minimizer can be obtained.