Constraining Earth's nonlinear mantle viscosity using plate-boundary resolving global inversions.

Variable viscosity in Earth's mantle exerts a fundamental control on mantle convection and plate tectonics, yet rigorously constraining the underlying parameters has remained a challenge. Inverse methods have not been sufficiently robust to handle the severe viscosity gradients and nonlinearities (arising from dislocation creep and plastic failure) while simultaneously resolving the megathrust and bending slabs globally. Using global plate motions as constraints, we overcome these challenges by combining a scalable nonlinear Stokes solver that resolves the key tectonic features with an adjoint-based Bayesian approach. Assuming plate cooling, variations in the thickness of continental lithosphere, slabs, and broad scale lower mantle structure as well as a constant grain size through the bulk of the upper mantle, a good fit to global plate motions is found with a nonlinear upper mantle stress exponent of 2.43 [Formula: see text] 0.25 (mean [Formula: see text] SD). A relatively low yield stress of 151 [Formula: see text] 19 MPa is required for slabs to bend during subduction and transmit a slab pull that generates asymmetrical subduction. The recovered long-term strength of megathrusts (plate interfaces) varies between different subduction zones, with South America having a larger strength and Vanuatu and Central America having lower values with important implications for the stresses driving megathrust earthquakes.

Changing Responses of PM2.5 and Ozone to Source Emissions in the Yangtze River Delta Using the Adjoint Model.

China's industrial restructuring and pollution controls have altered the contributions of individual sources to varying air quality over the past decade. We used the GEOS-Chem adjoint model and investigated the changing sensitivities of PM2.5 and ozone (O3) to multiple species and sources from 2010 to 2020 in the central Yangtze River Delta (YRDC), the largest economic region in China. Controlling primary particles and SO2 from industrial and residential sectors dominated PM2.5 decline, and reducing CO from multiple sources and ≥C3 alkenes from vehicles restrained O3. The chemical regime of O3 formation became less VOC-limited, attributable to continuous NOX abatement for specific sources, including power plants, industrial combustion, cement production, and off-road traffic. Regional transport was found to be increasingly influential on PM2.5. To further improve air quality, management of agricultural activities to reduce NH3 is essential for alleviating PM2.5 pollution, while controlling aromatics, alkenes, and alkanes from industry and gasoline vehicles is effective for O3. Reducing the level of NOX from nearby industrial combustion and transportation is helpful for both species. Our findings reveal the complexity of coordinating control of PM2.5 and O3 pollution in a fast-developing region and support science-based policymaking for other regions with similar air pollution problems.

Advances in Adjoint Functions of Connection Number in Water Resources Complex Systems: A Systematic Review.

The adjoint function of connection number has unique advantages in solving uncertainty problems of water resource complex systems, and has become an important frontier and research hotspot in the uncertainty research of water resource complex problems. However, in the rapid evolution of the adjoint function, some problems greatly limit the application of the adjoint function in the research of water resources. Therefore, based on bibliometric analysis, development, practical application issues, and prospects of the hot directions are analyzed. It is found that the development of the connection number of water resource set pair analysis can be divided into three stages: (1) relatively sluggish development before 2005, (2) a period of rapid advancement in adjoint function research spanning from 2005 to 2017, and (3) a subsequent surge post-2018. The introduction of the adjoint function of connection number promotes the continuous development of set pair analysis of water resources. Set pair potential and partial connection number are the crucial research directions of the adjoint function. Subtractive set pair potential has rapidly developed into a relatively independent and important trajectory. The research on connection entropy is comparatively less, which needs to be further strengthened, while that on adjacent connection number is even less. The adjoint function of set pair potential can be divided into three major categories: division set pair potential, exponential set pair potential, and subtraction set pair potential. The subtraction set pair potential, which retains the original dimension and quantity variation range of the connection number, is widely used in water resources and other fields. Coupled with the partial connection number, a series of new connection number adjoint functions have been developed. The partial connection number can be mainly divided into two categories: total partial connection number, and semi-partial connection number. Among these, the calculation expression and connotation of total partial connection numbers have not yet reached a consensus, accompanied by the slow development of high-order partial connection numbers. Semi-partial connection number can describe the mutual migration movement between different components of the connection number, which develops rapidly. With the limitations and current situation described above, promoting the exploration and application of the adjoint function of connection number in the field of water resources and other fields of complex systems has become the focus of future research.

Location-Specific Control of Precursor Emissions to Mitigate Photochemical Air Pollution.

The effects of precursor emission controls on air quality can vary greatly depending on where emission reductions occur. We use the adjoint of the Community Multiscale Air Quality (CMAQ) model to evaluate impacts of spatially targeted NOx emission reductions on odd oxygen (Ox = O3 + NO2). The air quality responses studied here include one population-weighted regionwide and three city-level receptors in Central California. We map high-priority locations for NOx control and their changes over decadal time scales. The desirability of NOx-focused emission control programs has increased between 2000 and 2022. We find for present-day conditions that reducing NOx emissions by 28% from targeted high-priority locations can achieve 60% of the air quality benefits of uniform NOx reductions at all locations. High-priority source locations are found to differ for individual city-level versus regionwide receptors of interest. While high-impact emission hotspots for improving city-level metrics are found within the city itself or closely adjacent, the spatial pattern of emission hotspots for improving regionwide air quality is more complex and requires comprehensive consideration of upwind sources. Results of this study can help to inform strategic decision-making at local and regional levels about where to prioritize emission control efforts.

Optimal control by deep learning techniques and its applications on epidemic models.

We represent the optimal control functions by neural networks and solve optimal control problems by deep learning techniques. Adjoint sensitivity analysis is applied to train the neural networks embedded in differential equations. This method can not only be applied in classic epidemic control problems, but also in epidemic forecasting, discovering unknown mechanisms, and the ideas behind can give new insights to traditional mathematical epidemiological problems.

An Inverse Modeling Approach to Estimate Three-Dimensional Aortic Valve Interstitial Cell Stress Fiber Force Levels.

Within the aortic valve (AV) leaflet exists a population of interstitial cells (AVICs) that maintain the constituent tissues by extracellular matrix (ECM) secretion, degradation, and remodeling. AVICs can transition from a quiescent, fibroblast-like phenotype to an activated, myofibroblast phenotype in response to growth or disease. AVIC dysfunction has been implicated in AV disease processes, yet our understanding of AVIC function remains quite limited. A major characteristic of the AVIC phenotype is its contractile state, driven by contractile forces generated by the underlying stress fibers (SF). However, direct assessment of the AVIC SF contractile state and structure within physiologically mimicking three-dimensional environments remains technically challenging, as the size of single SFs are below the resolution of light microscopy. Therefore, in the present study, we developed a three-dimensional (3D) computational approach of AVICs embedded in 3D hydrogels to estimate their SF local orientations and contractile forces. One challenge with this approach is that AVICs will remodel the hydrogel, so that the gel moduli will vary spatially. We thus utilized our previous approach (Khang et al. 2023, "Estimation of Aortic Valve Interstitial Cell-Induced 3D Remodeling of Poly (Ethylene Glycol) Hydrogel Environments Using an Inverse Finite Element Approach," Acta Biomater., 160, pp. 123-133) to define local hydrogel mechanical properties. The AVIC SF model incorporated known cytosol and nucleus mechanical behaviors, with the cell membrane assumed to be perfectly bonded to the surrounding hydrogel. The AVIC SFs were first modeled as locally unidirectional hyperelastic fibers with a contractile force component. An adjoint-based inverse modeling approach was developed to estimate local SF orientation and contractile force. Substantial heterogeneity in SF force and orientations were observed, with the greatest levels of SF alignment and contractile forces occurring in AVIC protrusions. The addition of a dispersed SF orientation to the modeling approach did not substantially alter these findings. To the best of our knowledge, we report the first fully 3D computational contractile cell models which can predict locally varying stress fiber orientation and contractile force levels.

Responses of Photochemical Air Pollution in California's San Joaquin Valley to Spatially and Temporally Resolved Changes in Precursor Emissions.

Ground-level ozone adversely affects human health and ecosystems. The effectiveness of control programs depends on which precursor(s) are controlled, by how much, and where and when emission reductions occur. We use the adjoint of the Community Multiscale Air Quality model to investigate odd oxygen (Ox ≡ O3 + NO2) sensitivities in California's San Joaquin Valley (SJV) to precursor emissions from local and upwind sources. Sensitivities are mapped and disaggregated by hour and day. Taken together, impacts of precursor emissions in the San Francisco Bay area and Sacramento Valley are similar in magnitude to impacts of local SJV emissions. Same-day emission sensitivities are mostly attributable to local sources, with the most influential anthropogenic emissions of VOCs (volatile organic compounds) and NOx (nitrogen oxides) occurring in the morning (9-11 am) and early afternoon hours (1-3 pm), respectively. For the northernmost SJV receptor, the influence from Sacramento Valley emissions peaks 5-6 h later than Bay area emissions; this difference diminishes for SJV receptors located further downwind. Results show a shift toward more NOx-sensitive conditions in the afternoon with all but the southernmost receptor shifting from VOC- to NOx-sensitive conditions. We also evaluate opportunities to control pollution through shifts in precursor emission location and timing.

Robust and efficient identification of optimal mixing perturbations using proxy multiscale measures.

Understanding and optimizing passive scalar mixing in a diffusive fluid flow at finite Péclet number [Formula: see text] (where [Formula: see text] and [Formula: see text] are characteristic velocity and length scales, and [Formula: see text] is the molecular diffusivisity of the scalar) is a fundamental problem of interest in many environmental and industrial flows. Particularly when [Formula: see text], identifying initial perturbations of given energy that optimally and thoroughly mix fluids of initially different properties can be computationally challenging. To address this challenge, we consider the identification of initial perturbations in an idealized two-dimensional flow on a torus that extremize various measures over finite time horizons. We identify such 'optimal' initial perturbations using the 'direct-adjoint looping' method, thus requiring the evolving flow to satisfy the governing equations and boundary conditions at all points in space and time. We demonstrate that minimizing multiscale measures commonly known as 'mix-norms' over short time horizons is a computationally efficient and robust way to identify initial perturbations that thoroughly mix layered scalar distributions over relatively long time horizons, provided the magnitude of the mix-norm's index is not too large. Minimization of such mix-norms triggers the development of coherent vortical flow structures which effectively mix, with the particular properties of these flow structures depending on [Formula: see text] and also the time horizon of interest. This article is part of the theme issue 'Mathematical problems in physical fluid dynamics (part 1)'.

A probabilistic approach to tomography and adjoint state methods, with an application to full waveform inversion in medical ultrasound.

Bayesian methods are a popular research direction for inverse problems. There are a variety of techniques available to solve Bayes' equation, each with their own strengths and limitations. Here, we discuss stochastic variational inference (SVI), which solves Bayes' equation using gradient-based methods. This is important for applications which are time-limited (e.g. medical tomography) or where solving the forward problem is expensive (e.g. adjoint methods). To evaluate the use of SVI in both these contexts, we apply it to ultrasound tomography of the brain using full-waveform inversion (FWI). FWI is a computationally expensive adjoint method for solving the ultrasound tomography inverse problem, and we demonstrate that SVI can be used to find a no-cost estimate of the pixel-wise variance of the sound-speed distribution using a mean-field Gaussian approximation. In other words, we show experimentally that it is possible to estimate the pixel-wise uncertainty of the sound-speed reconstruction using SVI and a common approximation which is already implicit in other types of iterative reconstruction. Uncertainty estimates have a variety of uses in adjoint methods and tomography. As an illustrative example, we focus on the use of uncertainty for image quality assessment. This application is not limiting; our variance estimator has effectively no computational cost and we expect that it will have applications in fields such as non-destructive testing or aircraft component design where uncertainties may not be routinely estimated.

Predicting Influenza Epidemic for United States.

Influenza causes repeat epidemics and huge loss of lives and properties. To predict influenza epidemics, we proposed an infectious disease dynamic prediction model with control variables (SEIR-CV), which considers the characteristics of the influenza epidemic transmission, seasonal impacts, and the intensity changes of control measures over time. The critical parameters of the model were inversed using an adjoint method. When using the surveillance data of the past 15 weeks to invert the parameters, the epidemic in the next 3 weeks in the United States can be accurately predicted. In addition, roll predictions from 26 September 2016 to 27 September 2018 were implemented. The correlation coefficient between the predicted values and the surveillance values was greater than 0.975, and the overall relative error of the predictions was less than 10%. These good model performances demonstrated the practicability and feasibility of SEIR-CV for influenza and corresponding infectious disease prediction.

A Framework for Four-Dimensional Variational Data Assimilation Based on Machine Learning.

The initial field has a crucial influence on numerical weather prediction (NWP). Data assimilation (DA) is a reliable method to obtain the initial field of the forecast model. At the same time, data are the carriers of information. Observational data are a concrete representation of information. DA is also the process of sorting observation data, during which entropy gradually decreases. Four-dimensional variational assimilation (4D-Var) is the most popular approach. However, due to the complexity of the physical model, the tangent linear and adjoint models, and other processes, the realization of a 4D-Var system is complicated, and the computational efficiency is expensive. Machine learning (ML) is a method of gaining simulation results by training a large amount of data. It achieves remarkable success in various applications, and operational NWP and DA are no exception. In this work, we synthesize insights and techniques from previous studies to design a pure data-driven 4D-Var implementation framework named ML-4DVAR based on the bilinear neural network (BNN). The framework replaces the traditional physical model with the BNN model for prediction. Moreover, it directly makes use of the ML model obtained from the simulation data to implement the primary process of 4D-Var, including the realization of the short-term forecast process and the tangent linear and adjoint models. We test a strong-constraint 4D-Var system with the Lorenz-96 model, and we compared the traditional 4D-Var system with ML-4DVAR. The experimental results demonstrate that the ML-4DVAR framework can achieve better assimilation results and significantly improve computational efficiency.

Efficient mesh refinement for the Poisson-Boltzmann equation with boundary elements.

The Poisson-Boltzmann equation is a widely used model to study electrostatics in molecular solvation. Its numerical solution using a boundary integral formulation requires a mesh on the molecular surface only, yielding accurate representations of the solute, which is usually a complicated geometry. Here, we utilize adjoint-based analyses to form two goal-oriented error estimates that allow us to determine the contribution of each discretization element (panel) to the numerical error in the solvation free energy. This information is useful to identify high-error panels to then refine them adaptively to find optimal surface meshes. We present results for spheres and real molecular geometries, and see that elements with large error tend to be in regions where there is a high electrostatic potential. We also find that even though both estimates predict different total errors, they have similar performance as part of an adaptive mesh refinement scheme. Our test cases suggest that the adaptive mesh refinement scheme is very effective, as we are able to reduce the error one order of magnitude by increasing the mesh size less than 20% and come out to be more efficient than uniform refinement when computing error estimations. This result sets the basis toward efficient automatic mesh refinement schemes that produce optimal meshes for solvation energy calculations.

Inverse design of the thermal environment in an airplane cockpit using the adjoint method with the momentum method.

Currently, the thermal environment in airplane cockpits is unsatisfactory and pilots often complain about a strong draft sensation in the cockpit. It is caused by the unreasonable air supply diffusers design. One of the best approaches to design a better cockpit environment is the adjoint method. The method can simultaneously and efficiently identify the number, size, location, and shape of air supply inlets, and the air supply parameters. However, the real air diffuser needed to design often have grilles, especially in the airplane cockpit, and the current method can only design the inlet as an opening. This study combined the adjoint method with the momentum method to directly identify the optimal air supply diffusers with grilles to create optimal thermal environment in an airplane cockpit (1) under ideal conditions and (2) with realistic constraints. Under the ideal conditions, the resulting design provides an optimal thermal environment for the cockpit, but it might not be feasible in practice. The design with realistic constraints provides acceptable thermal comfort in the cockpit, but it is not optimal. Thus, there is an engineering trade-off between design feasibility and optimization. All in all, the adjoint method with the momentum method can be effectively used to identify real air supply diffusers.

Expansion of the Nodal-Adjoint Method for Simple and Efficient Computation of the 2D Tomographic Imaging Jacobian Matrix.

This paper focuses on the construction of the Jacobian matrix required in tomographic reconstruction algorithms. In microwave tomography, computing the forward solutions during the iterative reconstruction process impacts the accuracy and computational efficiency. Towards this end, we have applied the discrete dipole approximation for the forward solutions with significant time savings. However, while we have discovered that the imaging problem configuration can dramatically impact the computation time required for the forward solver, it can be equally beneficial in constructing the Jacobian matrix calculated in iterative image reconstruction algorithms. Key to this implementation, we propose to use the same simulation grid for both the forward and imaging domain discretizations for the discrete dipole approximation solutions and report in detail the theoretical aspects for this localization. In this way, the computational cost of the nodal adjoint method decreases by several orders of magnitude. Our investigations show that this expansion is a significant enhancement compared to previous implementations and results in a rapid calculation of the Jacobian matrix with a high level of accuracy. The discrete dipole approximation and the newly efficient Jacobian matrices are effectively implemented to produce quantitative images of the simplified breast phantom from the microwave imaging system.

A new approach for health-oriented ozone control strategy: Adjoint-based optimization of NOx emission reductions using metaheuristic algorithms.

While levels of particulate matters in the Pearl River Delta Region (PRD) show a significant reduction, ozone (O3) has an opposite increasing trend, becoming the critical air quality target in this decade. Emission control strategies are typically formulated sector by sector, spatial variability in emissions reductions and health impacts of air pollutants may not be taken into account, affecting the overall effectiveness of control strategies. This study proposes an adjoint-based optimization framework to facilitate health-oriented O3 control over PRD. The location-specific adjoint sensitivity coefficients, which reflect the spatiotemporal influences from emissions of nitrogen dioxide (NOx) on O3 health impacts, are combined with metaheuristic algorithms to minimize the O3-related premature mortalities over receptor regions. Using the proposed optimization methodology, the regional O3 health benefits under current emission reduction policy can be increased by 16-27%. The results show that relatively larger NOx emissions reductions occurred at highly developed and populated areas. Particularly, significant reductions in NOx emissions are observed at Shenzhen and urban Guangzhou. Furthermore, implementing regional NOx emissions abatement has advantages to achieve an overall O3 health benefits for all cities. The interregional influences of NOx emissions abatement between cities indicate a promising strategy of health-oriented O3 control in PRD.

Atmospheric Warming Drives Growth in Arctic Sea Ice: A Key Role for Snow.

A number of feedbacks regulate the response of Arctic sea ice to local atmospheric warming. Using a realistic coupled ocean-sea ice model and its adjoint, we isolate a mechanism by which significant ice growth at the end of the melt season may occur as a lagged response to Arctic atmospheric warming. A series of perturbation simulations informed by adjoint model-derived sensitivity patterns reveal the enhanced ice growth to be accompanied by a reduction of snow thickness on the ice pack. Detailed analysis of ocean-ice-snow heat budgets confirms the essential role of the reduced snow thickness for persistence and delayed overshoot of ice growth. The underlying mechanism is a snow-melt-conductivity feedback, wherein atmosphere-driven snow melt leads to a larger conductive ocean heat loss through the overlying ice layer. Our results highlight the need for accurate observations of snow thickness to constrain climate models and to initialize sea ice forecasts.

Inverse design of an indoor environment using a filter-based topology method with experimental verification.

In order to create a healthy, comfortable, productive, and energy-efficient indoor environment, the computational fluid dynamics (CFD)-based adjoint method with an area-constrained topology method can be used to inversely design the optimal number, size, location, and shape of air supply inlets and air supply parameters. However, this method is not very mature, and the distribution of retained inlets is always scattered. To solve that problem, this investigation introduced a filter method that smooths the intermediate results during the inverse design process. Using a three-dimensional, non-isothermal, asymmetrical office with pre-set air supply inlets as an example, this study verified the performance of the proposed filter-based topology method. The verified method was then used to solve a multi-objective design problem and design an optimal indoor environment for a room. The results indicate that the proposed method was able to find the optimal number, location, and shape of air supply inlets and the optimal air supply temperature, velocity, and angle that led to a thermally comfortable, healthy, productive, and energy-efficient indoor environment. Finally, this investigation installed the optimal inlets in an environmental chamber to mimic the room. The measured air temperature, velocity, and mean age of air in several typical locations in the environmental chamber matched the CFD simulation results very closely.

Mapping ozone source-receptor relationship and apportioning the health impact in the Pearl River Delta region using adjoint sensitivity analysis.

While fine particulate matters are decreasing in the Pearl River Delta (PRD) region, the regional ozone (O3) shows an increasing trend that affects human health, leading to an urgent need for scientific understanding of source-receptor relationship between O3 and its precursor emissions given the changing background composition. We advanced and applied an adjoint air quality model to map contributions of individual O3 precursor emission sources [nitrogen oxides (NOx) and volatile organic compound (VOC)] at each location to annual regional O3 concentrations and to identify the possible dominant influential pathways of emission sources to O3 at different spatiotemporal scales. Additionally, we introduced the novel adjoint sensitivity approach to assess the relationship between precursor emissions and O3-induced premature mortality. Adjoint results show that Shenzhen was a major source contributor to regional O3 throughout all seasons, of which 49.4% (3.8%) were from its NOx (VOC) emissions. Local emissions (within PRD) contributed to 83% of the regional O3 whereas only ~54% of the estimated ~4000 regional O3-induced premature mortalities. The discrepancy between these two contributions was because O3-induced mortalities are dependent on not only O3 concentration, but incident rate and population density. We also found that a city with low O3-induced mortalities could have significant emission contributions to health impact in the region since the transport pathways could be through transport of local O3 or through transport of O3 precursors that form regional O3 thereafter. It is therefore necessary to formulate emission control policies from both air quality and public health perspectives, and it is also critical to have better understanding of influential pathways of emission sources to O3.

Global Optimization of Dielectric Metasurfaces Using a Physics-Driven Neural Network.

We present a global optimizer, based on a conditional generative neural network, which can output ensembles of highly efficient topology-optimized metasurfaces operating across a range of parameters. A key feature of the network is that it initially generates a distribution of devices that broadly samples the design space and then shifts and refines this distribution toward favorable design space regions over the course of optimization. Training is performed by calculating the forward and adjoint electromagnetic simulations of outputted devices and using the subsequent efficiency gradients for backpropagation. With metagratings operating across a range of wavelengths and angles as a model system, we show that devices produced from the trained generative network have efficiencies comparable to or better than the best devices produced by adjoint-based topology optimization, while requiring less computational cost. Our reframing of adjoint-based optimization to the training of a generative neural network applies generally to physical systems that can utilize gradients to improve performance.

Sensitivity Kernels of PP Precursor Traveltimes and Their Limitations for Imaging Topography of Discontinuities.

We analyze the sensitivity of PP precursor traveltimes that are often used to infer lateral variation in the depths of the 410- and 660-km discontinuities in the mantle. Previous results were inconclusive due to complex wave phenomena, such as multiple energy conversions and focusing/defocusing, that hamper their interpretation. Using spectral-element synthetics and Fréchet derivatives calculated with adjoint methods, we compute sensitivity kernels for volumetric and boundary parameters in a 1-D model for representative epicentral distances of past studies, and a dominant period of 11-25 s. Our results indicate that the boundary sensitivity of PP precursors is low and that these phases are not coherently seen in exact synthetics. Our most important finding is the strong sensitivity to both shear and compressional wave speeds, indicating that wave interference and wave conversions are dominant. The PP precursor traveltimes appear more sensitive to structural parameters, that is, compressional and shear wave speed, than to the boundaries; therefore, they are unlikely sources for valuable insight into discontinuity topography.