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This presentation provides an over of uncertainty in solar measurements from the National Laboratory of the Rockies' Solar Uncertainty Integrator (SUNI).
Simulation and control of many dynamic systems involve solving partial differential equations (PDE). This letter proposes a semi-analytical solution (SAS) approach for fast and high-quality solution of first-order PDEs. The region of interest of the studied PDE is divided into a grid, and an SAS is derived for each grid cell in the form of the multivariate polynomials, of which the coefficients are identified using initial value and boundary value conditions. The solutions are solved in a “time-stepping” manner, i.e. within one time step, the coefficients of the SAS are identified and the initial value of the next time step is evaluated. This approach achieves a significantly larger grid cell than the widely used finite difference method, and thus enhances the computational efficiency significantly. Furthermore, the simulation result on the natural gas pipeline model demonstrates the advantages of SAS in accuracy and computational efficiency.
In this work we explore the capability of physics-informed neural networks (PINNs) to discover multiple solutions. Many real-world phenomena governed by nonlinear differential equations (DEs), such as fluid flow, exhibit multiple solutions under the same conditions, yet capturing this solution multiplicity remains a significant challenge. A key difficulty lies in providing appropriate initial conditions or guesses, as widely used time-marching schemes and Newton’s method are highly sensitive to these choices when solving complex computational problems. While machine learning models, particularly PINNs, have shown promise in solving DEs, their ability to capture multiple solutions remains underexplored. In this work, we propose a simple and practical approach using PINNs to learn and discover multiple solutions. We first demonstrate that PINNs, when combined with random initialization and deep ensemble method—originally developed for uncertainty quantification—can effectively uncover multiple solutions to nonlinear ordinary and partial DEs. Although training large ensembles of PINNs may appear computationally demanding, this can be done efficiently using vectorization techniques supported by modern deep learning frameworks, allowing many networks to be trained simultaneously. Our approach highlights the critical role of initialization in shaping solution diversity, addressing an often-overlooked aspect of machine learning for scientific computing. Furthermore, we propose utilizing PINN-generated solutions as initial conditions or initial guesses for conventional numerical solvers to enhance accuracy and efficiency in capturing multiple solutions. Extensive numerical experiments, including the Allen–Cahn equation and cavity flow, where our approach successfully identifies both stable and unstable solutions, validate the effectiveness of our method. These findings establish a general and efficient framework for addressing solution multiplicity in nonlinear DEs.
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VENTSAR: Originated as VENTAX (Smith and Weber 1983) on the IBM Mainframe at the Savannah River Site (SRS) as a Fortran program. Updated to VENTSAR XL V1.0 (Simpkins 1997) as a spreadsheet version using macros created in Microsoft Excel. Later updated to VENTSAR XL V2.0 (Dixon 2018) as a spreadsheet executing a modified version of the original VENTAX Fortran code. Estimates contaminant concentrations on or near a building from a release at a nearby location. Calculates concentrations for a given meteorological exceedance probability or for a given stability and wind speed combination. Can model a single building with or without a penthouse on top or a ground location from either a stack or ground release. Plume rise can be considered. Contaminant releases can be chemical or radioactive with downwind concentrations determined at user-specified distances. Wind passing over and around buildings creates a complicated dispersion pattern. Air-intake vents may be located on building roofs or near the ground downwind of a release source. An estimation of pollutant concentrations on or near a structure is important in determining expected pollutant levels. Meteorological data are selected based on a specific area of the site. Fortran-based VENTAX able to make fast, complex mathematical calculations. Not user-friendly VENTSAR XL V1.0 no longer supported due to obsolete Excel macros. VENTSAR XL V2.0 an interim solution using an Excel spreadsheet to interface with the modified VENTAX Fortran code. Requires user to have Microsoft Excel. Not an intuitive interface for someone unfamiliar with VENTSAR. Does not perform dose calculations. VENTSAR V3.0 goal to combine the strengths of previous versions of VENTSAR into a single, powerful yet user-friendly program with a Graphical User Interface (GUI). Not dependent upon a specific program or plug-ins. Self-contained package to be used on any computer running Microsoft Windows. User-friendly, intuitive GUI created in Python for parameter input. Executes same modified VENTAX Fortran code as VENTSAR XL V2.0. Outputs formatted text file of completed calculations for easy review and dissemination. Performs inhalation and plume shine dose calculations for up to 11 user-selected nuclides from a nuclide dose factor library of almost 500 nuclides. 12 test cases were created for verification of V1.0 and V2.0. Same test cases run in V3.0 to verify correct performance. V3.0 produced similar results to V1.0. Confirmed GUI did not alter calculations in any way. Comparisons of the test case results confirm the V3.0 GUI does not influence the VENTSAR calculations. Simply passes same input parameters to VENTAX Fortran code for execution. Provides end user an easy-to-use, intuitive tool to quickly make building effect and plume rise calculations, as well as, inhalation and plume shine dose calculations. No experience with or working knowledge of Fortran, Excel, command line, or macros required. Self-contained package allows VENTSAR be deployed to any Windows computer without the need for specific programs or plug-ins.
Here, in this paper, we introduce general transfer operators between high-order and low-order refined finite element spaces that can be used to couple high-order and low-order simulations. Under natural restrictions on the low-order refined space we prove that both the high-to-low-order and low-to-high-order linear mappings are conservative, constant preserving, and high-order accurate. While the proof holds for affine geometries, numerical experiments indicate that the results hold for more general curved meshes. We present several numerical results confirming our analysis and demonstrate the utility of the new mappings in the context of adaptive mesh refinement and conservative multidiscretization coupling.
A large deformation, coupled finite-element (FE) model is developed to simulate the multiphase response of soft porous materials subjected to high strain-rate loading. The approach is based on the theory of porous media (TPM) at large deformations. Simplifications to the one-dimensional regime studied in the numerical simulations follow. An overview of several different time integration schemes is presented for the purpose of solving the nonlinear dynamic coupled balance of momenta (mixture and fluid) and balance of mass of the mixture equations. Numerical examples are presented for (i) verification against closed-form analytical solutions assuming small loads, (ii) demonstrating large deformation effects at high strain-rate, and (iii) showing differences in deformations between a single-phase elastodynamics model with occluded compressible pore fluid and a multiphase poroelastodynamics model at high strain-rate. The multiphase model shows that the relative motion of the pore fluid significantly dampens the deformation response of the solid skeleton as compared to the single-phase model, and makes it possible to extract quantitative values for the stresses of the different constituents, thereby allowing one to form preliminary conclusions about the onset of damage in the solid skeleton. The novelty of the current work is developing a multiphase, large deformation, mixture theory numerical model for high strain-rate loading of soft porous materials. It was discovered that explicit, adaptive time-stepping Runge–Kutta schemes offer high accuracy at relatively low cost when compared to traditional implicit or explicit central difference time-stepping schemes for shock-like loadings. Here, shock viscosity is added to the mixture momentum balance equation to regularize the shock front, and a stabilization term is added to the mixture mass balance equation to stabilize equal order interpolation finite elements for the coupled finite element solution of multiphase materials.
In this study, we provide several novel solutions of the coupled Ablowitz–Musslimani (AM) version of the nonlocal nonlinear Schrödinger (NLS) equation and the coupled nonlocal modified Korteweg–de Vries (mKdV) equation. In each case we compare and contrast the corresponding solutions of the relevant coupled local equations. Further, we provide new solutions of the coupled local NLS and coupled local mKdV equations which are not the solutions of the corresponding nonlocal equations. We also show that the nonlocal coupled (as well as uncoupled) mKdV equations have hidden Galilean invariance and admit novel solutions.
We propose an unsupervised machine-learning checkpoint-restart (CR) algorithm for particle-in-cell (PIC) algorithms using Gaussian mixtures (GM). The algorithm compresses the particle population per spatial cell by constructing a velocity distribution function using GM. Particles are reconstructed at restart time by local resampling of the Gaussians. To guarantee fidelity of the CR process, we ensure the exact preservation of invariants such as charge, momentum, and energy for both compression and reconstruction stages, everywhere on the mesh. We also ensure the preservation of Gauss' law after particle reconstruction by exactly matching the density profile at restart time. As a result, the GM CR algorithm is shown to provide a clean, conservative restart capability while potentially affording orders of magnitude savings in input/output requirements. Here, we demonstrate the algorithm using a recently developed exactly energy- and charge-conserving PIC algorithm using both electrostatic and electromagnetic tests. The tests demonstrate not only a high-fidelity CR capability, but also its potential for enhancing the fidelity of the PIC solution for a given particle resolution.
Here, the plane-stress assumption can be challenging to support in a finite element program because it traditionally requires separate implementations of constitutive models than those intended for three-dimensional or two-dimensional plane-strain simulations. As a solution to this issue, this paper presents a method to solve the plane-stress problem using a mixed formulation. In this formulation, the out-of-plane strain is treated as a field variable that is solved for in addition to the standard in-plane displacement variables, in a manner that weakly enforces the condition that the out-of-plane stress is zero. The proposed formulation is non-intrusive, requiring no modifications to the constitutive models in contrast to the conventional plane-stress formulation. The proposed mixed formulation has been benchmarked against analytical solutions and numerical solutions, with good performance and accuracy.
Here we present the analysis of approximation rates of operator learning in Chen and Chen (1995) and Lu et al. (2021), where continuous operators are approximated by a sum of products of branch and trunk networks. In this work, we consider the rates of learning solution operators from both linear and nonlinear advection–diffusion equations with or without reaction. We find that the approximation rates depend on the architecture of branch networks as well as the smoothness of inputs and outputs of solution operators.
Conservation voltage reduction (CVR) is implemented in the distribution system to reduce the voltage along the feeder with the objective to reduce the power consumption. Due to the increased penetration of distributed generators in the distribution system the centralized control is cumbersome. Hence, in this paper, a distributed algorithm is proposed for volt-var optimization (VVO) with CVR objective in a three-phase unbalanced distribution network. The proposed method is based on alternating direction method of multipliers (ADMM) algorithm. Mathematically the problem is formulated as a convex optimization problem with reactive power support from smart inverters. The convexity guarantees the solution obtained using the proposed method is a converged solution. The proposed method is validated on IEEE 4-bus and 13-bus three-phase unbalanced systems.
A large software team consisting of members with different expertise, skillsets, personalities, ethnicities, and involving collaboration on a large and complex software product presents many technical and cultural challenges, but also provides unique opportunities. In this article, we discuss the essential issues we faced when successfully transforming a collection of various independently developed software libraries into one large integrated product: the eXtreme-scale scientific Software Development Kit (xSDK). Furthermore, we argue it is just as important to pay attention to cultural challenges, such as establishment of reliable communication channels that considers, among others, differences in personalities and backgrounds as well as overcoming geographical separation and time-zone distribution when collaborating, as technical challenges. Finally, we discuss opportunities stemming from participating in a large diverse software team, such as increased internal expertise, variety of skillsets, broadened connections to external experts, and access to a larger pool of ideas or solutions.
Low-precision numeric types offer significant efficiency and energy benefits for computing applications. Mixed-precision algorithms, combining low and high precision types, maintain accuracy while improving performance. Despite advantages, there exist challenges on adapting existing mixed-precision algorithms to new technologies, such as new hardware architectures and new low-precision data types. This paper presents current challenges and opportunities to advance science in this domain targetting more energy efficient solutions.
The atomic structure and chemistry at metal/oxide interfaces play a crucial role in determining their properties. However, studying semi-coherent metal/oxide interfaces that include misfit dislocations through density functional theory (DFT) is often computationally expensive due to the large number of atoms involved, ranging from hundreds to thousands. In this study, we explore solute segregation behavior at the Fe/Y 2 O 3 interface—an important model interface for cladding applications in nuclear fission reactors—by combining DFT calculations with a machine learning (ML) approach. ML models are trained using DFT-calculated segregation energies (𝐸 𝑆𝑒𝑔 ) to identify the key chemical and geometric factors influencing solute segregation at metal/oxide interfaces, revealing the competition between these features in determining 𝐸 𝑆𝑒𝑔 . Moreover, the segregation behavior at a specific Fe/Y 2 O 3 interface is predicted with high accuracy using ML models trained on data from this interface. Furthermore, it is found that the ML models could also predict solute segregation at a different Fe/Y 2 O 3 interface with a new orientation relationship (OR), at a computational cost of less than 1/45 of that required for similar DFT calculations.
Deep learning (DL), in particular deep neural networks, by default is purely data-driven and in general does not require physics. This is the strength of DL but also one of its key limitations when applied to science and engineering problems in which underlying physical properties—such as stability, conservation, and positivity—and accuracy are required. DL methods in their original forms are often not capable of respecting the underlying mathematical models or achieving desired accuracy even in big-data regimes. On the other hand, many data-driven science and engineering problems, such as inverse problems, typically have limited experimental or observational data, and DL would overfit the data in this case. Leveraging information encoded in the underlying mathematical models, we argue, not only compensates for missing information in low data regimes but also provides opportunities to equip DL methods with the underlying physics, hence promoting better generalization. This paper develops a model-constrained DL approach and its variant TNet—a Tikhonov neural network—which are capable of learning not only information hidden in the training data but also in the underlying mathematical models to solve inverse problems governed by partial differential equations in low data regimes. We provide the constructions and some theoretical results for the proposed approaches for both linear and nonlinear inverse problems. Since TNet is designed to learn inverse solutions with Tikhonov regularization, it is interpretable: in fact it recovers Tikhonov solutions for linear cases while potentially approximating Tikhonov solutions for nonlinear inverse problems. We also prove that data randomization can enhance not only the smoothness of the networks but also their generalizations. Comprehensive numerical results confirm the theoretical findings and show that with even as little as 1 training data sample for one-dimensional (1D) deconvolution, 5 for an inverse 2D heat conductivity problem, 100 for inverse initial conditions for a time-dependent 2D Burgers’s equation, and 50 for inverse initial conditions for 2D Navier–Stokes equations, TNet solutions can be as accurate as Tikhonov solutions while being several orders of magnitude faster. Furthermore, this is possible owing to the model-constrained term, replications, and randomization.
Many imaging techniques for biological systems—like fixation of cells coupled with fluorescence microscopy—provide sharp spatial resolution in reporting locations of individuals at a single moment in time but also destroy the dynamics they intend to capture. In this study, these snapshot observations contain no information about individual trajectories, but still encode information about movement and demographic dynamics, especially when combined with a well-motivated biophysical model. The relationship between spatially evolving populations and single-moment representations of their collective locations is well-established with partial differential equations (PDEs) and their inverse problems. However, experimental data is commonly a set of locations whose number is insufficient to approximate a continuous-in-space PDE solution. Here, motivated by popular subcellular imaging data of gene expression, we embrace the stochastic nature of the data and investigate the mathematical foundations of parametrically inferring demographic rates from snapshots of particles undergoing birth, diffusion, and death in a nuclear or cellular domain. Toward inference, we rigorously derive a connection between individual particle paths and their presentation as a Poisson spatial process. Using this framework, we investigate the properties of the resulting inverse problem and study factors that affect quality of inference. One pervasive feature of this experimental regime is the presence of cell-to-cell heterogeneity. Rather than being a hindrance, we show that cell-to-cell geometric heterogeneity can increase the quality of inference on dynamics for certain parameter regimes. Altogether, the results serve as a basis for more detailed investigations of subcellular spatial patterns of RNA molecules and other stochastically evolving populations that can only be observed for single instants in their time evolution.
The latent variable proximal point (LVPP) algorithm is a framework for solving infinite-dimensional variational problems with pointwise inequality constraints. The algorithm is a saddle point reformulation of the Bregman proximal point algorithm. At the continuous level, the two formulations are equivalent, but the saddle point formulation is more amenable to discretization because it introduces a structure-preserving transformation between a latent function space and the feasible set. Working in this latent space is much more convenient for enforcing inequality constraints than the feasible set, as discretizations can employ general linear combinations of suitable basis functions, and nonlinear solvers can involve general additive updates. LVPP yields numerical methods with observed mesh-independence for obstacle problems, contact, fracture, plasticity, and others besides; in many cases, for the first time. The framework also extends to more complex constraints, providing means to enforce convexity in the Monge–Ampère equation and handling quasi-variational inequalities, where the underlying constraint depends implicitly on the unknown solution. Here, in this paper, we describe the LVPP algorithm in a general form and apply it to ten problems from across mathematics.