Solving equations using Khovanskii bases

We develop a new eigenvalue method for solving structured polynomial equations over any field. The equations are defined on a projective algebraic variety which admits a rational parameterization by a Khovanskii basis, e.g., a Grassmannian in its Plücker embedding. This generalizes established algorithms for toric varieties, and introduces the effective use of Khovanskii bases in computer algebra. We investigate regularity questions and discuss several applications

Discretize first, filter next: Learning divergence-consistent closure models for large-eddy simulation

We propose a new neural network based large eddy simulation framework for the incompressible Navier-Stokes equations based on the paradigm “discretize first, filter and close next”. This leads to full model-data consistency and allows for employing neural closure models in the same environment as where they have been trained. Since the LES discretization error is included in the learning process, the closure models can learn to account for the discretization. Furthermore, we employ a divergence-consistent discrete filter defined through face-averaging and provide novel theoretical and numerical filter analysis. This filter preserves the discrete divergence-free constraint by construction, unlike general discrete filters such as volume-averaging filters. We show that using a divergence-consistent LES formulation coupled with a convolutional neural closure model produces stable and accurate results for both a-priori and a-posteriori training, while a general (divergence-inconsistent) LES model requires a-posteriori training or other stability-enforcing measures

Energy-consistent discretization of viscous dissipation with application to natural convection flow

A new energy-consistent discretization of the viscous dissipation function in incompressible flows is proposed. It is implied by choosing a discretization of the diffusive terms and a discretization of the local kinetic energy equation and by requiring that continuous identities like the product rule are mimicked discretely. The proposed viscous dissipation function has a quadratic, strictly dissipative form, for both simplified (constant viscosity) stress tensors and general stress tensors. The proposed expression is not only useful in evaluating energy budgets in turbulent flows, but also in natural convection flows, where it appears in the internal energy equation and is responsible for viscous heating. The viscous dissipation function is such that a consistent total energy balance is obtained: the ‘implied’ presence as sink in the kinetic energy equation is exactly balanced by explicitly adding it as source term in the internal energy equation. Numerical experiments of Rayleigh–Bénard convection (RBC) and Rayleigh–Taylor instabilities confirm that with the proposed dissipation function, the energy exchange between kinetic and internal energy is exactly preserved. The experiments show furthermore that viscous dissipation does not affect the critical Rayleigh number at which instabilities form, but it does significantly impact the development of instabilities once they occur. Consequently, the value of the Nusselt number on the cold plate becomes larger than on the hot plate, with the difference increasing with increasing Gebhart number. Finally, 3D simulations of turbulent RBC show that energy balances are exactly satisfied even for very coarse grids. Therefore, the proposed discretization also forms an excellent starting point for testing sub-grid scale models and is a useful tool to assess energy budgets in any turbulence simulation, with or without the presence of natural convection

Assessing the development of internal disorders in pome fruit with X-ray CT before, during and after controlled atmosphere storage and shelf life

This study examined the use of X-ray computed tomography (CT) for the early detection of physiological disorders in pome fruit during controlled atmosphere (CA) storage and shelf life. The CT images of healthy and disordered ‘Braeburn’ apples, ‘Golden Delicious’ apples and ‘Conference’ pears were evaluated. ‘Braeburn’ apple (n = 80) were scanned with CT before and during browning-inducing CA conditions (0.5 °C, 1.5 kPa O2, 5 kPa CO2) and subsequent shelf life. ‘Conference’ pears (n = 70) were scanned following regular air storage that induced freezing injury (−2 °C) and additional CA storage (−0.6 °C, 3 kPa O2, <0.7 kPa CO2) and subsequent shelf life. ‘Golden Delicious’ apples (n = 60) were scanned after CA storage (1 °C, 1 kPa O2, 3 kPa CO2) and after 35 days of shelf life. The causes of postharvest losses after CA storage and shelf life were core browning, flesh browning and bitter pit for ‘Braeburn’, ‘Conference’ and ‘Golden Delicious’, respectively. After CA storage, the mean greyscale value (MGV) of CT images was higher in healthy ‘Braeburn’ and ‘Golden Delicious’ apples compared to those that appeared externally healthy but later developed a disorder during shelf life. The MGV decreased during storage and shelf life in affected ‘Braeburn’ and ‘Golden Delicious’ apples, whereas no change occurred during shelf life for healthy ‘Golden Delicious’ apples, and ‘Braeburn’ apples stored for 17 weeks. No difference in the MGV was found between healthy and disordered ‘Conference’ pears. For ‘Braeburn’, voids associated with core browning did not develop until fruit were removed from CA storage and subsequently kept in shelf life for at least seven days. Results of this study indicate that the MGV of CT images can be used to indicate ‘Braeburn’ and ‘Golden Delicious’ apple fruit marketability after CA storage before shelf life

Scientific machine learning: A symbiosis

This editorial serves as a preface to the "Scientific Machine Learning" (SciML) special issue of the AIMS Foundations of Data Science journal. In this piece, we contend that SciML exists in a symbiotic relationship with the fields of computational science and engineering (CSE) and machine learning (ML). We highlight the progress (and limitations) of CSE and reflect on the recent successes of ML. While ML creates significant possibilities for advancing simulation techniques, it lacks the mathematical guarantees that are typically found in CSE. We argue that as SciML develops and embraces the remarkable capabilities of ML, it will support, not replace, traditional methods of CSE. We then overview some existing challenges and opportunities in this interdisciplinary field and close by introducing the special issue papers

Analysis and formal specification of OpenJDK's BitSet: Proof files

This artifact [1] (accompanying our iFM 2023 paper [2]) describes the software we developed that contributed towards our analysis of OpenJDK's BitSet class. This class represents a vector of bits that grows as needed. Our analysis exposed numerous bugs. In our paper, we proposed and compared a number of solutions supported by formal specifications. Full mechanical verification of the BitSet class is not yet possible due to limited support for bitwise operations in KeY and bugs in BitSet. Our artifact contains proofs for a subset of the methods and new proof rules to support bitwise operators

ComPEQ-MR: Compressed Point Cloud dataset with Eye Tracking and Quality assessment in Mixed Reality

Point clouds (PCs) have attracted researchers and developers due to their ability to provide immersive experiences with six degrees of freedom (6DoF). However, there are still several open issues in understanding the Quality of Experience (QoE) and visual attention of end users while experiencing 6DoF volumetric videos. First, encoding and decoding point clouds require a significant amount of both time and computational resources. Second, QoE prediction models for dynamic point clouds in 6DoF have not yet been developed due to the lack of visual quality databases. Third, visual attention in 6DoF is hardly explored, which impedes research into more sophisticated approaches for adaptive streaming of dynamic point clouds. In this work, we provide an open-source Compressed Point cloud dataset with Eye-tracking and Quality assessment in Mixed Reality (ComPEQ - MR). The dataset comprises four compressed dynamic point clouds processed by Moving Picture Experts Group (MPEG) reference tools (i.e., VPCC and GPCC), each with 12 distortion levels. We also conducted subjective tests to assess the quality of the compressed point clouds with different levels of distortion. The rating scores are attached to ComPEQ - MR so that they can be used to develop QoE prediction models in the context of MR environments. Additionally, eye-tracking data for visual saliency is included in this dataset, which is necessary to predict where people look when watching 3D videos in MR experiences. We collected opinion scores and eye-tracking data from 41 participants, resulting in 2132 responses and 164 visual attention maps in total. The dataset is available at https://ftp.itec.aau.at/datasets/ComPEQ-MR/

A qubit, a coin, and an advice string walk into a relational problem

Relational problems (those with many possible valid outputs) are different from decision problems, but it is easy to forget just how different. This paper initiates the study of FBQP/qpoly, the class of relational problems solvable in quantum polynomial-Time with the help of polynomial-sized quantum advice, along with its analogues for deterministic and randomized computation (FP, FBPP) and advice (/poly, /rpoly). Our first result is that FBQP/qpoly/= FBQP/poly, unconditionally, with no oracle - a striking contrast with what we know about the analogous decision classes. The proof repurposes the separation between quantum and classical one-way communication complexities due to Bar-Yossef, Jayram, and Kerenidis. We discuss how this separation raises the prospect of near-Term experiments to demonstrate "quantum information supremacy," a form of quantum supremacy that would not depend on unproved complexity assumptions. Our second result is that FBPP/ FP/poly - that is, Adleman s Theorem fails for relational problems - unless PSPACE NP/poly. Our proof uses IP = PSPACE and time-bounded Kolmogorov complexity. On the other hand, we show that proving FBPP/FP/poly will be hard, as it implies a superpolynomial circuit lower bound for PromiseBPEXP. We prove the following further results: Unconditionally, FP/= FBPP and FP/poly/= FBPP/poly (even when these classes are carefully defined). FBPP/poly = FBPP/rpoly (and likewise for FBQP). For sampling problems, by contrast, SampBPP/poly/= SampBPP/rpoly (and likewise for SampBQP)

Energy-conserving neural network for turbulence closure modeling

In turbulence modeling, we are concerned with finding closure models that represent the effect of the subgrid scales on the resolved scales. Recent approaches gravitate towards machine learning techniques to construct such models. However, the stability of machine-learned closure models and their abidance by physical structure (e.g. symmetries, conservation laws) are still open problems. To tackle both issues, we take the ‘discretize first, filter next’ approach. In this approach we apply a spatial averaging filter to existing fine-grid discretizations. The main novelty is that we introduce an additional set of equations which dynamically model the energy of the subgrid scales. Having an estimate of the energy of the subgrid scales, we can use the concept of energy conservation to derive stability. The subgrid energy containing variables is determined via a data-driven technique. The closure model is used to model the interaction between the filtered quantities and the subgrid energy. Therefore the total energy should be conserved. Abiding by this conservation law yields guaranteed stability of the system. In this work, we propose a novel skew-symmetric convolutional neural network architecture that satisfies this law. The result is that stability is guaranteed, independent of the weights and biases of the network. Importantly, as our framework allows for energy exchange between resolved and subgrid scales it can model backscatter. To model dissipative systems (e.g. viscous flows), the framework is extended with a diffusive component. The introduced neural network architecture is constructed such that it also satisfies momentum conservation. We apply the new methodology to both the viscous Burgers' equation and the Korteweg-De Vries equation in 1D. The novel architecture displays superior stability properties when compared to a vanilla convolutional neural network