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Deep ReLU networks have exponential expressive potential but behave identically to their shallow counterparts under random initialization. This paper introduces a novel training paradigm that constrains weights into mathematically optimized patterns. This enables the resulting network to make exponential use of its depth.
A legacy FORTRAN 77 implementation of the Baer–Nunziato two-phase mixture theory for deflagration-to-detonation transition (DDT) in reactive granular materials—hereafter the XDDT (eXplosive DDT) code—has been modularized to Fortran 90 with modular structure, external input files, and adaptive mesh capability. During validation, two code defects were identified and corrected: an inconsistency in the nodal solid pressure evaluation and a nonphysical burn-front tracking criterion. The ignition criterion was also corrected to use the granular surface temperature from the interface heat transfer model, matching the original Baer implementation. An initial attempt to validate against Figure 3 of the original Baer and Nunziato (1986) paper revealed that the code’s detonation velocity on a 201-node mesh (5.5 km/s) was approximately 21% below the expected Chapman–Jouguet value for 70% TMD HMX (∼7 km/s). Validation was redirected to the piston-driven DDT experiments of McAfee et al. (1989), Shot B-9036, for which well-characterized ionization-pin data are available. With the compaction-burn coefficient calibrated to 𝐶 𝛼 = 75, the XDDT code reproduces the DDT transition time to within 0.4% and produces a steady-state detonation velocity within 4% of the McAfee experimental value of 6.36 km/s. The burn model was generalized to support pressure-dependent exponents, enabling application to nitrocellulose-based ball propellants (TS3659) with a cube-root pressure dependence. Validation against the Sandusky/Baer PDC82 piston-impact experiment yielded a reactive wave velocity of 2.3–2.8 km/s, in good agreement with the experimental value of ∼2.2 km/s, and wave coalescence within 5% of the experimental timing. The mathematical model, input parameter requirements, and a roadmap for extending XDDT to PETN with an autocatalytic burn model are presented.
Version 2.0 of the RESRAD-RDD&IND computer code is designed to support the implementation of protective action guides (PAGs) after a nuclear emergency incident including a radiological dispersal device (RDD) and/or an improvised nuclear device (IND) incident (EPA 2017). Eight different group types, addressing various decisions, are available for selection. The RESRAD-RDD&IND code calculates radiological doses, stay times, etc., for the selected group that the user wishes to focus on. (That is, the results for all the groups are not calculated simultaneously, and the input for those other groups do not matter, although some parameter values are shared between groups.) Version 2.0 has a user-friendly interface so that the RESRAD-RDD&IND code can be used with minimal training. For example, the user can select the major characteristics of the problem-event type, source term, and decision type from the left side of the interface and then calculate the results with the default assumptions for the exposure scenarios. More in-depth analysis would include specifying site-specific exposure scenario characteristics in the right side of the interface. The procedures for data entry and results viewing are self-explanatory. This is because common window maneuvering features and text instructions were incorporated in the interface design. General and context-specific help are available to aid users entering parameter values, as well. The RESRAD-RDD&IND computer code gives the user the option to select either an RDD or IND incident for analysis. For an RDD event analysis, 11 radionuclides (Am-241, Cf-252, Cm-244, Co-60, Cs-137, Ir-192, Po-210, Pu-238, Pu-239, Ra-226, and Sr-90) are included. These 11 radionuclides are the radionuclides most likely used for an RDD. More than 90 radionuclides can be selected for an IND event analysis. Initial default concentrations are provided for 44 radionuclides for a uranium-fueled IND event. These 44 radionuclides are those that would contribute significantly to the radiation dose associated with a uranium-fueled bomb detonation. The radionuclides generated from ingrowth of these 44 initial radionuclides are also automatically included in the analysis. Pu-239, Cs-134m, Ru-105, and Rb-89 and their progeny can be selected for analysis if they are detected and their concentrations are determined. This user’s guide, which is Volume 2 of the User’s Manual for RESRAD-RDD&IND Code Version 2, provides instructions to users on how to install the RESRAD-RDD&IND code, navigate the interface, and use the various features, including those discussed above, to set up an analysis and view/print the results in text outputs. Volume 1 of the User’s Manual for RESRAD-RDD&IND Code Version 2 (Yu et al. 2026), which contains descriptions of the methodology and theoretical basis for dose modeling and the mathematical equations implemented in the code, can be accessed and viewed through the Help menu in the code or can be downloaded from the RESRAD website (https://resrad.evs.anl.gov).
Variational Calculus is an advanced topic in mathematics. This document exists to compile the basics of Variational Calculus in a readable and digestible manner for application by engineers. This document is a living document and will be updated as time moves forward. It is the hope that by reading this document, the basics of Variational Calculus can be readily applied to any system/function of interest. This does not serve as a replacement for textbooks or other learned sources but should act as a companion piece to better serve the reader in times of confusion and distress.
The characterization of Hamiltonians and other components of open quantum dynamical systems plays a crucial role in quantum computing and other applications. Scientific machine learning techniques have been applied to this problem in a variety of ways, including by modeling with deep neural networks. However, the majority of mathematical models describing open quantum systems are linear, and the natural nonlinearities in learnable models have not been incorporated using physical principles. We present a data-driven model for open quantum systems that includes learnable, thermodynamically consistent terms. The trained model is interpretable, as it directly estimates the system Hamiltonian and linear components of coupling to the environment. We validate the model on synthetic two and three-level data, as well as experimental two-level data collected from a quantum device at Lawrence Livermore National Laboratory.
Data generated in studies of cellular regulatory systems are often qualitative. For example, measurements of signaling readouts in the presence and absence of mutations may reveal a rank ordering of responses across conditions but not the precise extents of mutation-induced differences. Qualitative data are often ignored by mathematical modelers or are considered in an ad hoc manner, as in the study of Kocieniewski and Lipniacki (2013) [Phys Biol 10: 035006], which was focused on the roles of MEK isoforms in ERK activation. In this earlier study, model parameter values were tuned manually to obtain consistency with a combination of qualitative and quantitative data. This approach is not reproducible, nor does it provide insights into parametric or prediction uncertainties. Here, starting from the same data and the same ordinary differential equation (ODE) model structure, we generate formalized statements of qualitative observations, making these observations more reusable, and we improve the model parameterization procedure by applying a systematic and automated approach enabled by the software package PyBioNetFit. We also demonstrate uncertainty quantification (UQ), which was absent in the original study. Our results show that PyBioNetFit enables qualitative data to be leveraged, together with quantitative data, in parameterization of systems biology models and facilitates UQ. These capabilities are important for reliable estimation of model parameters and model analyses in studies of cellular regulatory systems and reproducibility.
Emerging tensor network techniques for solutions of partial differential equations (PDEs), known for their ability to break the curse of dimensionality, deliver new mathematical methods for ultra-fast numerical solutions of high-dimensional problems. Here, we introduce a Tensor Train (TT) Chebyshev spectral collocation method, in both space and time, for the solution of the time-dependent convection-diffusion-reaction (CDR) equation with inhomogeneous boundary conditions, in Cartesian geometry. Previous methods for numerical solution of time-dependent PDEs often used finite difference for time, and a spectral scheme for the spatial dimensions, which led to a slow linear convergence. Spectral collocation space-time methods show exponential convergence; however, for realistic problems they need to solve large four-dimensional systems. We overcome this difficulty by using a TT approach, as its complexity only grows linearly with the number of dimensions. We show that our TT space-time Chebyshev spectral collocation method converges exponentially, when the solution of the CDR is smooth, and demonstrate that it leads to a very high compression of linear operators from terabytes to kilobytes in TT-format, and a speedup of tens of thousands of times when compared to a full-grid space-time spectral method. These advantages allow us to obtain the solutions at much higher resolutions.
The global magnificent four theory is the homological version of a maximally supersymmetric $(8+1)$-dimensional gauge theory on a Calabi-Yau fourfold fibered over a circle. In the case of a toric fourfold we conjecture the formula for its twisted Witten index. String-theoretically we count the BPS states of a system of $D0$-$D2$-$D4$-$D6$-$D8$-branes on the Calabi-Yau fourfold in the presence of a large Neveu-Schwarz $B$-field. Mathematically, we develop the equivariant $K$-theoretic DT4 theory, by constructing the four-valent vertex with generic plane partition asymptotics. Physically, the vertex is a supersymmetric localization of a non-commutative gauge theory in $8+1$ dimensions.
In this study, development of compact photocurrent models is currently dominated by analytical techniques that rely on physical assumptions to render the governing equations solvable in a closed form. Violation of these assumptions can reduce the accuracy of the models and/or limit their scope. In this paper we show that system identification of nonlinear state-space systems can serve as an alternative numerical basis for non-intrusive reduced order modeling of photocurrent effects. To that end we develop a compact gray box photocurrent model (GBPM) by using a state-space representation with a low-dimensional latent state equation that mimics a mathematical model for the response of an idealized class of devices to ionizing radiation. In so doing we obtain a model that learns the dynamics of a quantity of interest directly from its measurements without requiring snapshots of the internal device state or its discretized model, and can be inferred from very small data sets. To demonstrate the approach we train the GBPM using a small experimental data set for a Z5236 Zener diode and a small synthetic data set obtained by simulating a synthetic pn-junction device. We then compare the GBPMs with black box models trained on the same data and show that performance of the latter is limited by the size of the data set, while the former are able to achieve excellent performance in both the reproductive and the predictive regimes.
We investigate MHV tree-level gravity amplitudes as defined on the spinor-helicity variety. Unlike their gluon counterparts, the gravity amplitudes do not have logarithmic singularities and do not admit Amplituhedron-like construction. Importantly, they are not determined just by their singularities, but rather their numerators have interesting zeroes. We make a conjecture about the uniqueness of the numerator and explore this feature from a more mathematical perspective. This leads us to a new approach for examining adjoints. We outline steps of our proposed proof and provide computational evidence for its validity in specific cases.
In this article we review, for a mathematical audience, the computation of (tree-level) scattering amplitudes in Yang-Mills theory in detail. In particular we demonstrate explicitly how the same formulas for six-particle NMHV helicity amplitudes are obtained from summing Feynman diagrams and from computing the canonical form of the n=6, k=1, m=4 amplituhedron.
Learning operators between infinitely dimensional spaces is an important learning task arising in machine learning, imaging science, mathematical modeling and simulations, etc. This paper studies the nonparametric estimation of Lipschitz operators using deep neural networks. Non-asymptotic upper bounds are derived for the generalization error of the empirical risk minimizer over a properly chosen network class. Under the assumption that the target operator exhibits a low dimensional structure, our error bounds decay as the training sample size increases, with an attractive fast rate depending on the intrinsic dimension in our estimation. Our assumptions cover most scenarios in real applications and our results give rise to fast rates by exploiting low dimensional structures of data in operator estimation. We also investigate the influence of network structures (e.g., network width, depth, and sparsity) on the generalization error of the neural network estimator and propose a general suggestion on the choice of network structures to maximize the learning efficiency quantitatively.
Weak form Scientific Machine Learning (WSciML) is a recently developed framework for data-driven modeling and scientific discovery. It leverages the weak form of equation error residuals to provide enhanced noise robustness in system identification via convolving model equations with test functions, reformulating the problem to avoid direct differentiation of data. The performance, however, relies on wisely choosing a set of compactly supported test functions. In this work, we mathematically motivate a novel data-driven method for constructing Single-scale-Local reference functions for creating the set of test functions. Our approach numerically approximates the integration error introduced by the quadrature and identifies the support size for which the error is minimal, without requiring access to the model parameter values. Through numerical experiments across various models, noise levels, and temporal resolutions, we demonstrate that the selected supports consistently align with regions of minimal parameter estimation error. We also compare the proposed method against the strategy for constructing Multi-scale-Global (and orthogonal) test functions introduced in our prior work, demonstrating the improved computational efficiency.
Stochastic programming provides a rigorous mathematical framework for making decisions under uncertainty in a risk-aware manner. Two-stage stochastic programming is, perhaps, the simplest form of this framework. Here the first-stage variables represent decisions that must be made "here and now" in the face of uncertainty, while the second-stage variables are decisions made after uncertain events. However, the broad adoption of stochastic programming has been hindered by computational challenges caused by the two-stage stochastic programming formulation which requires solving an ensemble of optimization problems. Using quantum amplitude estimation (QAE), quantum computers have shown the theoretic ability to compute expectations with Monte-Carlo methods with quadratically fewer samples than classical methods. In this work, we present a quantum algorithm for computing the expectation term using QAE for given first-stage decisions. Further, we detail methods of computing gradient information from the quantum calculation enabling the application of classical gradient-based optimization techniques. The result is a classical-quantum hybrid method of solving two-stage stochastic programs. These techniques are demonstrated with computational experiments based an engineering optimization problem.
This study focuses on design parameter sensitivity studies pertaining to a steam generator (SG) model, using both Python and machine-learning tools. The SG model is a mathematical representation (including fluid flow and heat transfer equations/models/correlations) of a steam-generating unit in a pressurized water reactor (PWR)-type small modular reactor (SMR) system. Design studies involve changing the model’s input design parameters (e.g., temperature, pressure, mass flow rate) to observe the resulting effects on the output of the system (e.g., heat transfer coefficient [HTC], Nusselt number, heat transfer performance). Sensitivity studies analyze the degree to which system output and/or desired parameters (e.g., HTC or heat transfer performance) are sensitive to changes in input parameters. By using machine-learning tools such as the Risk Analysis Virtual Environment (RAVEN) developed at Idaho National Laboratory (INL), detailed design parametric sensitivity studies and model optimization were performed. Six input parameters—namely, the pressure, temperature, and mass flow rate for the inlet of the primary-side (hot fluid) and secondary-side (cold fluid) of the SG—were randomly perturbed via RAVEN’s Monte Carlo Sampler module, using uniform distributions (±1% relative changes). The analysis results give valuable insights into SG system performance and optimization, and provide justification for researching optimized sensor placement to effectively monitor and obtain experimental data.
This study focuses on design parameter sensitivity studies pertaining to a steam generator (SG) model, using both Python and machine-learning tools. The SG model is a mathematical representation (including fluid flow and heat transfer equations/models/correlations) of a steam-generating unit in a pressurized water reactor (PWR)-type small modular reactor (SMR) system. Design studies involve changing the model’s input design parameters (e.g., temperature, pressure, mass flow rate) to observe the resulting effects on the output of the system (e.g., heat transfer coefficient [HTC], Nusselt number, heat transfer performance). Sensitivity studies analyze the degree to which system output and/or desired parameters (e.g., HTC or heat transfer performance) are sensitive to changes in input parameters. By using machine-learning tools such as the Risk Analysis Virtual Environment (RAVEN) developed at Idaho National Laboratory (INL), detailed design parametric sensitivity studies and model optimization were performed. Six input parameters—namely, the pressure, temperature, and mass flow rate for the inlet of the primary-side (hot fluid) and secondary-side (cold fluid) of the SG—were randomly perturbed via RAVEN’s Monte Carlo Sampler module, using uniform distributions (±1% relative changes). The analysis results give valuable insights into SG system performance and optimization, and provide justification for researching optimized sensor placement to effectively monitor and obtain experimental data.
Master's Thesis for an MA in Mathematics at Arizona State University titled "Data Integration Methods for Bayesian Inverse Problems."
Automation control is a key strategy to improve the economic competitiveness of nuclear power plants. Not only does it help reduce operational costs, but it also extends the value proposition of these plants to nontraditional markets, including unattended operations in remote villages and space. However, the dynamics of the operating environments of nuclear reactors are subject to changes over time, and there are no widely adopted methods to ensure that the automation strategy will remain effective over the extended durations required for these applications. Adaptive control is a discipline that offers the possibility to accommodate such changes online. However, it relies on mathematical assumptions that must be respected to ensure robustness and reliability. In this work, we derive an adaptive control formulation for linear systems in which all states are observable and apply it to an instance of load-follow operation for Boiling Water Reactors. We assumed uncertainty in two factors: the temperature coefficient and the control rod worth, both of which are affected over time by the evolution of the nuclear reactor core environment. With an arbitrary penalty factor of 5, we found that the mean absolute and integral time absolute errors can be reduced by more than 90%, underscoring the strength of adaptive control. To extend the application to more challenges, different uncertainties and load-follow trajectories, as well as new formulations that include non-linearity and partial observability, are currently being developed.