A scalable framework for learning the geometry-dependent solution operators of partial differential equations
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Yin, Minglang (ORCID:000000017861350X)↗
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Results for “DIFFERENTIAL OPERATOR”
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Analysis into Asymptotic Convergence to Full Nonlinear Solutions and Exploration of the Implication of Numerical Operator Mutation of Differential Systems
A robust, sufficiently accurate and practical hydrodynamic simulation toolset is required as a key component of the modeling and simulation of air-gap electrostatic discharge events. This work was performed to complement these ongoing efforts. In particular, hydrodynamic simulations must be vetted to ensure they are robust and sufficiently accurate over relevant characteristic scales. Verification models were generated in order to cultivate the technical knowledge and expertise needed to properly create, implement and execute numerical simulations. Furthermore, this effort was utilized extensively to educate students on the mathematical and numerical principles underlying hydrodynamic simulations. This education opportunity, provided in a holistic and rigorous manner, has greatly benefited developing scientists and engineers with the necessary understandings and toolsets required to excel at accomplishing the task at hand, and, more generally, it has enabled them to generate key programmatic deliverables. This report articulates several subtilties; specifically, how perturbations, nonlinear behavior, and dissipative mechanisms influence numerical stability, how to properly structure mathematical and numerical solutions, and how to properly generate error estimation/assignment. A more rigorous discussion of the consequences of such topics can be found in the body of this report in Chapters 2 and 3 with qualitative findings discussed in Chapter 4.
97 MATHEMATICS AND COMPUTING↗
Deep transfer operator learning for partial differential equations under conditional shift
Transfer learning enables the transfer of knowledge gained while learning to perform one task (source) to a related but different task (target), hence addressing the expense of data acquisition and labelling, potential computational power limitations and dataset distribution mismatches. Here, we propose a new transfer learning framework for task-specific learning (functional regression in partial differential equations) under conditional shift based on the deep operator network (DeepONet). Task-specific operator learning is accomplished by fine-tuning task-specific layers of the target DeepONet using a hybrid loss function that allows for the matching of individual target samples while also preserving the global properties of the conditional distribution of the target data. Inspired by conditional embedding operator theory, we minimize the statistical distance between labelled target data and the surrogate prediction on unlabelled target data by embedding conditional distributions onto a reproducing kernel Hilbert space. We demonstrate the advantages of our approach for various transfer learning scenarios involving nonlinear partial differential equations under diverse conditions due to shifts in the geometric domain and model dynamics. Our transfer learning framework enables fast and efficient learning of heterogeneous tasks despite considerable differences between the source and target domains.
42 ENGINEERING↗
Cycloidal wave energy converter using fixed differential buoyancy to control operating torques
A floating mooring system for a single CycWEC applies counter forces and torques to keep a generator suitably stationary for power generation without requiring fixed attachments to the ocean floor or requiring a large frame interconnecting multiple CycWECs. The mooring system uses floats or floatation structure with differential ballasting to counter operating torque and drag plates to counter reactive forces. The floatation structures may be used to float the CycWEC for transport to a deployment location, where changing the overall ballasting of the floatation structures submerges the CycWEC to a desired depth and differential ballasting in the floatation structures counts expected operating torques.
Siegel, Stefan G.↗
Learning Only on Boundaries: A Physics-Informed Neural Operator for Solving Parametric Partial Differential Equations in Complex Geometries
Recently, deep learning surrogates and neural operators have shown promise in solving partial differential equations (PDEs). However, they often require a large amount of training data and are limited to bounded domains. In this work, we present a novel physics-informed neural operator method to solve parameterized boundary value problems without labeled data. By reformulating the PDEs into boundary integral equations (BIEs), we can train the operator network solely on the boundary of the domain. This approach reduces the number of required sample points from $O(N^d)$ to $O(N^{d-1}$), where $d$ is the domain’s dimension, leading to a significant acceleration of the training process. Additionally, our method can handle unbounded problems, which are unattainable for existing physics-informed neural networks (PINNs) and neural operators. Finally, our numerical experiments show the effectiveness of parameterized complex geometries and unbounded problems.
97 MATHEMATICS AND COMPUTING↗
Learning the solution operator of parametric partial differential equations with physics-informed DeepONets
Partial differential equations (PDEs) play a central role in the mathematical analysis and modeling of complex dynamic processes across all corners of science and engineering. Their solution often requires laborious analytical or computational tools, associated with a cost that is markedly amplified when different scenarios need to be investigated, for example, corresponding to different initial or boundary conditions, different inputs, etc. In this work, we introduce physics-informed DeepONets, a deep learning framework for learning the solution operator of arbitrary PDEs, even in the absence of any paired input-output training data. We illustrate the effectiveness of the proposed framework in rapidly predicting the solution of various types of parametric PDEs up to three orders of magnitude faster compared to conventional PDE solvers, setting a previously unexplored paradigm for modeling and simulation of nonlinear and nonequilibrium processes in science and engineering.
97 MATHEMATICS AND COMPUTING↗
Resummation of the Holstein-Primakoff expansion and differential equation approach to operator square roots
Not Available
75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗
Linear differential
Apparatuses and methods of operating a linear differential (100, 600) are described herein. The linear differential (100, 600) contains a slide portion (102) with parallel right-hand and left-hand threaded rods (112, 114). Threaded onto the right-hand and left-hand threaded rods (112, 114) and attached to the slide portion (102) are right-hand and left-hand gears (116, 118). Meshed between the right-hand and left-hand gears (116, 118) and also attached to the slide portion (102) is a driven gear (200). An end effector (104) is attached to the driven gear (200) and is configured to translate along a translation axis (110) and rotate around a rotation axis (120).
Crawford, Anthony↗
A review of peridynamic theory and nonlocal operators along with their computer implementations
Not Available
97 MATHEMATICS AND COMPUTING↗
Importance of hydrogen oxidation reaction current in quantifying hydrogen crossover in PEM water electrolyzers at high differential pressure
Understanding hydrogen permeation in proton exchange membrane water electrolyzers (PEMWEs) operating at high differential pressures (>25 bar) is critical towards developing effective gas recombination strategies that enable safe operation and high efficiency. Developing this understanding relies on accurate quantification of hydrogen crossover rates in water electrolyzers operating under such conditions. In this work, we show that PEMWEs operating at high differential pressures exhibit noticeable hydrogen oxidation reaction (HOR) currents. As the HOR consumes part of the permeated hydrogen at the anode, neglecting HOR currents leads to severe underestimation of the hydrogen crossover rate. We implemented a new method combining hydrogen oxidation current with online gas chromatography measurements to accurately quantify hydrogen crossover rates as a function of operating current density in PEMWEs operating at high differential pressures (10–30 bar g ).
25 ENERGY STORAGE↗
Linking structure to performance: Characterization of porous transport layers for high-pressure water electrolysis
Proton exchange membrane (PEM) water electrolysis is a promising technology to produce cost-efficient hydrogen. PEM electrolyzers offer a large current density range and the ability to operate at differential pressure which can be used to minimize both capital and operational expenditures. However, directly producing pressurized hydrogen at the cathode results in pushing the membrane against the anode porous transport layer (PTL). This can lead to detrimental effects, such as membrane deformation or ruptures, which depend on membrane properties as well as PTL material properties such as pore size, structure, and morphology. In this work, a range of sinter and felt-based commercial PTLs are evaluated for their contributions to the cell's electrochemical and H 2 crossover performance at cathode pressures up to 30 bar. X-ray tomography and post-operando optical microscopy are used to assess the morphology of the PTLs, and the PTL induced deformation experienced by the catalyst coated membrane (CCM), respectively. PTL samples with lower porosity were found to reduce both the cell voltage and the amount of H 2 permeating from the cathode to the anode exhaust, which was ascribed to improved catalyst layer contact and reduced membrane deformation, respectively. The best performing PTLs improved electrolyzer efficiency by ~1.5 kWh/kg H2 . Specifically, 1 kWh/kg H2 was gained due to reducing membrane deformation and decreasing H 2 crossover. The remainder 0.5 kWh/kg H2 were achieved by improving the electrical contact at the electrode/PTL interface which decreased cell voltage.
08 HYDROGEN↗
Operator inference with roll outs for learning reduced models from scarce and low-quality data
Data-driven modeling has become a key building block in computational science and engineering. However, data that are available in science and engineering are typically scarce, often polluted with noise and affected by measurement errors and other perturbations, which makes learning the dynamics of systems challenging. Here, in this work, we propose to combine data-driven modeling via operator inference with the dynamic training via roll outs of neural ordinary differential equations. Operator inference with roll outs inherits interpretability, scalability, and structure preservation of traditional operator inference while leveraging the dynamic training via roll outs over multiple time steps to increase stability and robustness for learning from low-quality and noisy data. Numerical experiments with data describing shallow water waves and surface quasi-geostrophic dynamics demonstrate that operator inference with roll outs provides predictive models from training trajectories even if data are sampled sparsely in time and polluted with noise of up to 10%.
97 MATHEMATICS AND COMPUTING↗
Using X-ray radiography to study oxygen flow in a proton exchange membrane electrolyzer operating under balanced pressure conditions
Of the various water electrolyzer technologies, the proton exchange membrane electrolyzer (PEMWE) is one of the best solutions for producing clean hydrogen without releasing CO 2 . In order to allow for widespread use of clean hydrogen, it is necessary to decrease its cost, which is intrinsically related to system operation. Current PEMWE plants operate in differential mode, directly pressurizing hydrogen and benefiting from thermodynamic compression, which increases overall system efficiency. However, high differential pressure above 30 bar can cause membrane stress, resulting in membrane creeping and failure. Pressurizing the water and operating at balanced pressure allows hydrogen to be produced at higher pressures while preserving the integrity of the membrane and porous layers. Nevertheless, the impact of pressurizing water on PEMWE performance must be better understood to maximize performance under balanced pressure conditions. Here, this study examined the impact of water pressure on two-phase flow. A high-pressure electrolyzer setup was developed to perform operando X-ray radiography and examine oxygen transport with high temporal resolution. The imaging segmentation process, developed to capture bubble properties in the channel, was applied to a specific experiment. The results clearly showed that as pressure increased up to 30 bars, the initial bubbly flow transitioned to slug flow, which led to channel saturation with oxygen. This work demonstrates that two-phase flow in an electrolyzer can be studied using X-ray radiography, which has the advantages of fast measurements and the ability to probe dense materials, such as those required for pressurized electrolyzers.
Balanced high pressure operation↗
Active operator learning with predictive uncertainty quantification for partial differential equations
With the increased prevalence of neural operators being used to provide rapid solutions to partial differential equations (PDEs), understanding the accuracy of model predictions and the associated error levels is necessary for deploying reliable surrogate models in scientific applications. Existing uncertainty quantification (UQ) frameworks employ ensembles or Bayesian methods, which can incur substantial computational costs during both training and inference. Here, we propose a lightweight predictive UQ method tailored for Deep operator networks (DeepONets) that also generalizes to other operator networks. Numerical experiments on linear and nonlinear PDEs demonstrate that the framework’s uncertainty estimates are unbiased and provide accurate out-of-distribution uncertainty predictions with a sufficiently large training dataset. Our framework provides fast inference and uncertainty estimates that can efficiently drive outer-loop analyses that would be prohibitively expensive with conventional solvers. We demonstrate how predictive uncertainties can be used in the context of Bayesian optimization and active learning problems to yield improvements in accuracy and data-efficiency for outer-loop optimization procedures. In the active learning setup, we extend the framework to Fourier Neural Operators (FNO) and describe a generalized method for other operator networks. To enable real-time deployment, we introduce an inference strategy based on precomputed trunk outputs and a sparse placement matrix, reducing evaluation time by more than a factor of five. Our method provides a practical route to uncertainty-aware operator learning in time-sensitive settings.
97 MATHEMATICS AND COMPUTING↗
Neural operator transformers capture bifurcating drift-wave turbulence in fusion plasma simulations
Self-consistent modeling of turbulence-driven transport is critical for optimizing confinement in magnetically confined fusion plasmas, such as tokamaks and stellarators. In particular, capturing the long-term co-evolution of turbulence, flow, and background plasma profiles remains computationally challenging. Direct numerical simulation of these multiscale, highly nonlinear processes is often demanding and impractical for real-time control or design optimization. To address this bottleneck, we investigate transformer-based neural operator partial differential equation surrogates for emulating the dynamics of drift-wave turbulence bifurcation mediated by zonal flows, using the modified Hasegawa–Wakatani (MHW) model as a prototypical system. We find that the finetuned neural operator model has excellent performance in capturing the multi-spatiotemporal-scales of MHW turbulence bifurcation and is robust to testing on rare and out-of-distribution dynamics. Specifically, we demonstrate that a single unified model accurately predicts both quasi-steady-state turbulence and a wide range of dynamical transition processes, such as nonlinear saturation, spontaneous suppression of turbulence, and the emergence of macroscopic zonal flows, over time horizons vastly exceeding the local turbulence correlation time. This computationally efficient approach establishes a strong foundation for fast, AI-based modeling of complex, multiscale phenomena in magnetized fusion plasmas.
70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
Design of digital acquisition for beam current monitor
As a part of the Proton Improvement Plan – II (PIP-II) at Fermilab, instrumentation systems are being modernized to take advantage of the higher speeds and ease of use offered by standardized embedded systems like MicroTCA. A rear-transition module (RTM) is being designed to interface with said embedded systems. In each of the four identical channels on the RTM, the differential signal from an alternating-current current transformer (ACCT) transimpedance amplifier will again be amplified by a differential operation-amplifier, then filtered by a low-pass topology. The conditioned signal is then digitized at a maximum of 10MS/s by an analog to digital converter (ADC) integrated circuit. After digitization, the ADC passes the data to an off the shelf AdvancedMC (AMC) Xilinx FPGA module using low voltage differential signals. This paper will describe the simulation of analog circuitry for signal conditioning, simulation of digital signal integrity based on physical design as well as verification of design characteristics critical to signal integrity. This work aims to create a methodology that can be applied to future RTMs requiring application of high-speed digital design principles.
White, R.Turner [Fermilab]↗
Design of Digital Acquisition for Beam Current Monitor
As a part of the Proton Improvement Plan II (PIP-II) at Fermilab, instrumentation systems are being modernized to take advantage of the higher speeds and ease of use offered by standardized embedded systems like MicroTCA. A rear-transition module (RTM) is being designed to interface with said embedded systems. In each of the four identical channels on the RTM, the differential signal from an alternating-current current transformer (ACCT) transimpedance amplifier will again be amplified by a differential operation-amplifier, then filtered by a low-pass topology. The conditioned signal is then digitized at a maximum of 10MS/s by an analog to digital converter (ADC) integrated circuit. After digitization, the ADC passes the data to an off the shelf AdvancedMC (AMC) Xilinx FPGA module using low voltage differential signals. This paper will describe the simulation of analog circuitry for signal conditioning, simulation of digital signal integrity based on physical design as well as verification of design characteristics critical to signal integrity. This work aims to create a methodology that can be applied to future RTMs requiring application of high-speed digital design principles.
White, Turner [Fermilab] (ORCID:0009000348159256)↗
Data-driven linear time advance operators for the acceleration of plasma physics simulation
In this study, we demonstrate the application of data-driven linear operator construction for time advance with a goal of accelerating plasma physics simulation. We apply dynamic mode decomposition (DMD) to data produced by the nonlinear SOLPS-ITER (Scrape-off Layer Plasma Simulator - International Thermonuclear Experimental Reactor) plasma boundary code suite in order to estimate a series of linear operators and monitor their predictive accuracy via online error analysis. We find that this approach defines when these dynamics can be represented by a sequence of approximate linear operators and is essential for providing consistent projections when compared to an unconstrained application. For linear diffusion and advection–diffusion fluid test problems, we construct and apply operators within explicit and implicit time advance schemes, demonstrating that stability can be robustly guaranteed in each case. We further investigate the use of the linear time advance operators within several integration methods including forward Euler, backward Euler, and the matrix exponential. The application of this method to simulation data from SOLPS-ITER, with varying levels of Markov chain Monte Carlo numerical noise, shows that constrained DMD operators yield a capability to identify, extract, and integrate a (slow) subset of the present timescales. Example applications show that for projected speedup factors of [Formula: see text], and [Formula: see text], a mean relative error of 3%, 5%, and 8% and maximum relative error less than 20% are achievable, which appears acceptable for typical SOLPS-ITER steady-state simulations.
70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗