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At least 289 records · Page 16

Real-time estimators for scattering observables: A full account of finite-volume errors for quantum simulation

The real-time correlators of quantum field theories can be directly probed through new approaches to simulation, such as quantum computing and tensor networks. This provides a new framework for computing scattering observables in lattice formulations of strongly interacting theories, such as lattice quantum chromodynamics. In this paper, we prove that the proposal of real-time estimators of scattering observables is universally applicable to all scattering observables of gapped quantum field theories. All finite-volume errors are exponentially suppressed, and the rate of this suppression is controlled by the regulator considered, namely, a displacement of the spectrum of the theory into the complex plane. A partial restoration of Lorentz symmetry by averaging over different boosts gives an additional suppression of finite volume errors. Our results also apply to the simulation of wave packet scattering, where a similar averaging is performed to construct the wave packets that regulate the finite volume effects. This result represents a necessary key step toward determining a broad class of scattering observables via quantum computing that are currently inaccessible via classical computing. Such observables are relevant for various applications, including hadron spectroscopy, hadron structure, and precision tests of the Standard Model. We also comment on potential applications of our results to traditional computational schemes.

Burbano, Ivan M. [University of California, Berkel↗

Analog and symbolic computation through the Koopman framework

We develop a Koopman operator framework for studying the computational structure of dynamical systems. Specifically, we show that the resolvent of the Koopman operator provides a natural abstraction of halting, yielding a ‘Koopman halting problem’ that is recursively enumerable in general. For symbolic systems, such as those defined on Cantor space, this operator formulation captures reachability between clopen sets, while for equicontinuous systems we prove that the Koopman halting problem is decidable. Our framework demonstrates that absorbing (halting) states in coarse-grained finite automata correspond to Koopman eigenfunctions with eigenvalue one, while cycles in the transition graph impose spectral constraints associated with periodic dynamics. These results provide a unifying perspective on computation in symbolic and analog systems, showing how computational universality is reflected in operator spectra, invariant subspaces, and algebraic structures. Beyond symbolic dynamics, this operator-theoretic lens opens pathways to analyze the computational properties of a broader class of dynamical systems, including polynomial and analog models, and suggests that computational hardness may admit dynamical signatures in terms of Koopman spectral structure.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

Agentic framework for programmatic crystal structure generation using a fine-tuned worker–supervisor large language model

Platinum group metals (PGMs) underpin many catalytic technologies but face severe supply constraints, motivating the search for alternative materials and computational methods to accelerate discovery. While atomistic simulation tools such as Pymatgen and ASE have streamlined structure manipulation, they require detailed inputs, limiting accessibility for experimentalists and slowing early-stage exploration. Here, in this study, we present an AI-driven agentic framework that orchestrates worker–supervisor large language models (LLMs). The worker translates natural-language prompts of varying abstraction into valid crystallographic structures using a compact LLM fine-tuned with low-rank adaptation on a curated text–code–CIF dataset, emphasizing energy-efficient training. Benchmarking against the baseline CodeGen-350M-mono model shows that fine-tuning reduces hallucination rates from 100% to as low as 5% and improves structural match accuracy to up to 82% for fully specified inputs. Accuracy declines with decreasing prompt detail but remains nontrivial even when only stoichiometry and space group are provided, underscoring the LLM’s capacity for crystallographic inference. The supervisor Claude LLM evaluates the outputs and triggers iterative refinement through the worker’s built-in structure manipulation capabilities (e.g., supercell scaling, strain, vacancy, and substitution operations). We further demonstrate use cases for technologically relevant catalysts, including IrO 2 , pyrochlore Pb 2 Ir 2 O 7 , Ni 2 FeO 4 , and Ni 3 Mo, where the framework generates physically consistent structures that can be refined via geometry optimization. This work introduces a low-energy, language-driven pathway for integrating human and machine intelligence in materials design, paving the way for AI-assisted synthesis planning and high-throughput screening of complex oxides.

AI agent↗

Emulator-based Bayesian calibration of a subglacial drainage model

Subglacial drainage models, often motivated by the relationship between hydrology and ice flow, sensitively depend on numerous unconstrained parameters. We explore using borehole water-pressure time series to calibrate the uncertain parameters of a popular subglacial drainage model, taking a Bayesian perspective to quantify the uncertainty in parameter estimates and in the calibrated model predictions. To reduce the computation time associated with Markov Chain Monte Carlo sampling, we construct a fast Gaussian process emulator to stand in for the subglacial drainage model. We first carry out a calibration experiment using synthetic observations consisting of model simulations with hidden parameter values as a demonstration of the method. Using real borehole water pressures measured in western Greenland, we find meaningful constraints on four of the eight model parameters and a factor-of-three reduction in uncertainty of the calibrated model predictions. These experiments illustrate Gaussian process-based Bayesian inference as a useful tool for calibration and uncertainty quantification of complex glaciological models using field data. However, significant differences between the calibrated model and the borehole data suggest that structural limitations of the model, rather than poorly constrained parameters or computational cost, remain the most important constraint on subglacial drainage modelling.

58 GEOSCIENCES↗

More Than 30 000-fold Field Enhancement of Terahertz Nanoresonators Enabled by Rapid Inverse Design

The rapid development of 6G communications using terahertz (THz) electromagnetic waves has created a demand for highly sensitive THz nanoresonators capable of detecting these waves. Among the potential candidates, THz nanogap loop arrays show promising characteristics but require significant computational resources for accurate simulation. This requirement arises because their unit cells are 10 times smaller than millimeter wavelengths, with nanogap regions that are 1 000 000 times smaller. To address this challenge, we propose a rapid inverse design method using physics-informed machine learning, employing double deep Q-learning with an analytical model of the THz nanogap loop array. In ~39 h on a middle-level personal computer, our approach identifies the optimal structure through 200 000 iterations, achieving an experimental electric field enhancement of 32 000 at 0.2 THz, 300% stronger than prior results. In conclusion, our analytical model-based approach significantly reduces the amount of computational resources required, offering a practical alternative to numerical simulation-based inverse design for THz nanodevices.

21 terahertz time-domain spectroscopy↗

Deep-learning atomistic semi-empirical pseudopotential model for nanomaterials

The semi-empirical pseudopotential method (SEPM) has been widely applied to provide computational insights into the electronic structure, photophysics, and charge carrier dynamics of nanoscale materials. We present “DeepPseudopot”, a machine-learned atomistic pseudopotential model that extends the SEPM framework by combining a flexible neural network representation of the local pseudopotential with parameterized non-local and spin-orbit coupling terms. Trained on bulk quasiparticle band structures and deformation potentials from GW calculations, the model captures many-body and relativistic effects with very high accuracy across diverse semiconducting materials, as illustrated for silicon and group III-V semiconductors. DeepPseudopot’s accuracy, efficiency, and transferability make it well-suited for data-driven in silico design and discovery of novel optoelectronic nanomaterials.

Lin, Kailai [University of California, Berkeley, C↗

Deep operator network surrogate for phase-field modeling of metal grain growth during solidification

A deep operator network (DeepONet) has been constructed that generates accurate representations of phase-field model simulations for evolving two dimensional metal grain morphology growing from melt. These representations serve as lower resolution, computationally efficient stand-ins for quick parameter space exploration of solutions to the the Allen-Cahn equations that dictate the phase-field model simulations. The experimental target for the phase-field model is a uranium casting system cooling a 434 g uranium charge from a maximum temperature of 1400° C at an average rate of 30° C / min , traversing the crystallographic phases of the pure metal. Experimental parameters inform the phase-field model, whose higher resolution computational model solutions are used to train the DeepONet in a given parameter space with the aim of developing a faster, more efficient method for predicting the solidifying metal's microstructure at different potential experimental values. The final DeepONet generates high accuracy, lower resolution predictions with cumulative relative approximation error over all timesteps of less than 0.5%, while ensuring solutions remain within physically feasible ranges. Further, these relative error values are comparable with other state-of-the-art DeepONet models for microstructure evolution, while significantly reducing the amount of training data required. Training a convolutional neural network simultaneously with the DeepONet, enforcing realistic values at the complex metal grain boundaries, and mathematically encoding boundary conditions into the structure of the DeepONet improved prediction accuracy and computational efficiency over a standard DeepONet model.

36 MATERIALS SCIENCE↗

Deep-freeze graph training for latent learning

Scientific and engineering advances are primarily driven by multi-tier conceptual constructs and conditional theoretical frameworks. The theories allow predictions of hypothetical system responses, given a set of approximate conditions (ranges of applicability) imposed on latent parameters that cannot be measured directly. Learning to estimate the latent variables (Latent Learning) helps to pinpoint the anticipated range-edge anomalies and improves the confidence in interpretation, interpolation and extrapolation of limited experimental data. Due to high dimensionality and extreme non-linearity of the materials science problems, very large datasets are typically required for conventional data-driven model development. The vital experimental data collection, particularly on microstructural phases, is very challenging, which makes it difficult to compile a high-quality database. Incorporation of the domain knowledge into the computational graph structure, initialization and optimization processes presents a viable mechanism for developing accurate models, with limited datasets. Furthermore, this study successfully utilized the approach to build the Deep Freeze Graph (DeepFreG) by mapping known causality relationships and by digitizing empirical domain knowledge for Latent Learning (LL), with specific applications in materials science.

36 MATERIALS SCIENCE↗

System, method, and computer program for creating an internal conforming structure

A system for creating an internal formation of a tubular structure having an inner surface via additive manufacturing. The system broadly includes a computer modeling system and an additive manufacturing system. The computer modeling system may include a processor for generating a lattice cellular component via computer-aided design software according to inputs received from a user. The processor may also generate an internal formation lattice structure based on the lattice cellular component and modify the lattice structure to follow and/or conform to the curvature of the inner surface of the outer wall of the tubular structure. The additive manufacturing system may be configured to produce the lattice structure and the tubular structure via additive manufacturing material deposited layer by layer according to the lattice structure.

42 ENGINEERING↗

Chemical classification program synthesis using generative artificial intelligence

Accurately classifying chemical structures is essential for cheminformatics and bioinformatics, including tasks such as identifying bioactive compounds of interest, screening molecules for toxicity to humans, finding non-organic compounds with desirable material properties, or organizing large chemical libraries for drug discovery or environmental monitoring. However, manual classification is labor-intensive and difficult to scale to large chemical databases. Existing automated approaches either rely on manually constructed classification rules, or are deep learning methods that lack explainability. This work presents an approach that uses generative artificial intelligence to automatically write chemical classifier programs for classes in the Chemical Entities of Biological Interest (ChEBI) database. These programs can be used for efficient deterministic run-time classification of SMILES structures, with natural language explanations. The programs themselves constitute an explainable computable ontological model of chemical class nomenclature, which we call the ChEBI Chemical Class Program Ontology (C3PO). We validated our approach against the ChEBI database, and compared our results against deep learning models and a naive SMARTS pattern based classifier. C3PO outperforms the naive classifier, but does not reach the performance of state of the art deep learning methods. However, C3PO has a number of strengths that complement deep learning methods, including explainability and reduced data dependence. C3PO can be used alongside deep learning classifiers to provide an explanation of the classification, where both methods agree. The programs can be used as part of the ontology development process, and iteratively refined by expert human curators.

Artificial Intelligence↗

Exploring the Neutron Substructure with Advanced Polarized Helium-3 Targets (Or: How I Learned to Stop Worrying and Love Spectroscopy)

As we seek to understand the smallest, physical aspects of our universe, we cannot simply rely on our senses to probe the world around us as we did in the past. The smallest physical elements of our universe behave in strange, probabilistic ways and are completely invisible to the naked eye/ear/etc. So, we design clever experiments (such as scattering experiments) to probe these minute realms. Then, just as with the larger, observable world, we devise models and equations to describe what we think is happening. Due to the nature of the physical universe at the quantum scale and with the aid of symmetries such as Lorentz invariance, we can write down equations that describe the scattering, but the expressions contain functions, which we call ?form factors? and ?structure functions?, that we cannot compute from first principles. We can, however, formulate models that make predictions for these functions. By comparing our predictions with the observed data, we can gain insight into the validity of our models and thus a better physical understanding of what is happening at these minuscule scales. Studying the constituents inside of the nucleus of an atom adds another layer of difficulty if we can?t remove those components from the nucleus. This is the case with the neutron. When not bound in the nucleus with protons and other neutrons, the neutron will decay into a proton after about 15 minutes. So, we?re forced to study the neutron while it is still bound in the nucleus of an atom such as helium-3 (3He). For the last 1,000 years (rounding up), our group has developed high quality, polarized 3He targets made of an aluminosilicate glass. These targets are made in order to perform experiments at Jefferson Lab (JLab), experiments which let us determine the form factors and structure functions of the neutron by scattering polarized electrons from polarized neutrons (or rather polarized 3He). The specific experiments reported on in this thesis push the bounds of our understanding of the internal structure of the neutron. Good science is often about pushing experimental techniques to a new level. Toward that goal we study our polarized 3He targets both to advance the technology and to choose the best ones for our experiments. We do this using a process called nuclear magnetic resonance (NMR) to gauge the maximum polarization of a target and how fast the polarization decays with time. While these tests primarily provide us information that make analysis of our experimental scattering data possible, they also let us determine whether or not a target-cell is useful or even, dare I say, of spectacular quality. Our latest targets utilize a novel convection design allowing 3He to be polarized and quickly moved in front of the electron-beam, making it possible to use larger targets with higher electron-beam currents than ever before. This means more electrons scatter and we get more data. And by studying our targets in detail prior to using them in our experiments, we have found techniques to take effects which could have been detrimental to target quality and turn them to our advantage! It?s a real case of making lemonade out of lemons. We also use laser spectroscopy to study the absorption lines of alkali-metals in the target (potassium and rubidium, specifically). We add these alkali-metals to our target to facilitate polarizing the 3He. We can use the measurement of these pressure broadened absorption lines to determine the 3He density inside of the target with great precision. Historically, we understood the width of these lines would be dependent on the temperature of the target. Specifically, if I raise the temperature, the width should get bigger. I found that was not the case, which was very confusing at first, though very exciting now that I realize the data are self-consistent and suggestive of unexpected behavior. This thesis details the development of high quality, glass, polarized 3He targets for the 2020 An 1 /dn 2 and 2023 Gn E experiments, which utilized the first 3He convection targets and broke records in target quality. This thesis also covers the initial development of metal windows for the next-generation of 3He target-cells. Finally, this thesis documents the temperature dependence of the width of potassium (K) and rubidium (Rb) absorption lines as measured with laser spectroscopy.

Jantzi, Christopher↗

Knowledge-matching based computational framework for genome-scale metabolic model refinement

Genome-scale metabolic models (GEMs) are mathematically structured knowledge base reconstructed from annotated genome of different organisms. With the advancement of next-generation sequencing technology, many organisms have had their genomes sequenced. However, obtaining a high-quality GEM is highly time-consuming, even with the introduction of several genome-scale reconstruction tools that offer automated draft network generation and gap filling. It has been recognized that the iterative process of manual curation and refinement is the limiting step of GEM development, and how to expedite the GEM refinement is still an open question. As cellular metabolism is a complex system with very high degree of freedom and redundancy, the principles and techniques developed in process systems engineering can be adapted to expedite GEM refinement. In this paper we present a knowledge-matching based computation framework for GEM refinement, and demonstrate the effectiveness of the proposed solution using the refinement of a GEM for Clostridium tyrobutyricum.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

Detecting hidden layers from spreading dynamics on complex networks

When dealing with spreading processes on networks it can be of the utmost importance to test the reliability of data and identify potential unobserved spreading paths. In this paper we address these problems and propose methods for hidden layer identification and reconstruction. We also explore the interplay between difficulty of the task and the structure of the multilayer network describing the whole system where the spreading process occurs. Our methods stem from an exact expression for the likelihood of a cascade in the susceptible-infected model on an arbitrary graph. We then show that by imploring statistical properties of unimodal distributions and simple heuristics describing joint likelihood of a series of cascades one can obtain an estimate of both existence of a hidden layer and its content with success rates far exceeding those of a null model. Furthermore, we conduct our analyses on both synthetic and real-world networks providing evidence for the viability of the approach presented.

97 MATHEMATICS AND COMPUTING↗

Develop and verify soil/structure interaction for pile/foundation interaction

Phase II of the Offshore Code Comparison Collaboration, Continued, with Correlation and unCertainty (OC6) project was used to verify the implementation of a new soil-structure interaction (SSI) model for use within offshore wind turbine modeling software. The REDWIN Macro-element model implemented and verified in this study enables a computationally efficient way to model the linear and nonlinear SSI problem, including hysteretic damping, of a monopile structure. The modeling approach was integrated into several modeling tools and a series of increasingly complex simulations was conducted using the IEA 10MW reference turbine mounted on a monopile support structure to verify the coupling between the tools and the REDWIN Macro-element SSI model. This campaign includes only numerical verification between various software and modeling approaches so no experimental measurements are available. The load cases (LC) considered include: LC1 – static response of the tower and substructure LC2 – frequency and mode-shape analysis of the tower and substructure LC3 – response of the tower and substructure due to wind-only loading LC4 – response of the tower and substructure due to wave-only loading LC5 – response of the tower and substructure due to wind and wave loading. Detailed properties of the modeled system are found in the following reference, “Bergua, Roger, Amy Robertson, Jason Jonkman, and Andy Platt. 2021. "Specification Document for OC6 Phase II: Verification of an Advanced Soil-Structure Interaction Model for Offshore Wind Turbines.” Golden, CO: National Renewable Energy Laboratory. NREL/TP-5000-79938. https://www.nlr.gov/docs/fy21osti/79938.pdf. Details on the results from the OC6 Phase II project can be found in the following reference, “Bergua R, Robertson A, Jonkman J, et al. OC6 Phase II: Integration and verification of a new soil–structure interaction model for offshore wind design.” Wind Energy. 2022;25(5):793-810. doi:10.1002/we.2698

17 WIND ENERGY↗

Benchmarking the PCMCI Causal Discovery Algorithm for Spatiotemporal Systems

Causal discovery algorithms construct hypothesized causal graphs that depict causal dependencies among variables in observational data. While powerful, the accuracy of these algorithms is highly sensitive to the underlying dynamics of the system in ways that have not been fully characterized in the literature. In this report, we benchmark the PCMCI causal discovery algorithm in its application to gridded spatiotemporal systems. Effectively computing grid-level causal graphs on large grids will enable analysis of the causal impacts of transient and mobile spatial phenomena in large systems, such as the Earth’s climate. We evaluate the performance of PCMCI with a set of structural causal models, using simulated spatial vector autoregressive processes in one- and two-dimensions. We develop computational and analytical tools for characterizing these processes and their associated causal graphs. Our findings suggest that direct application of PCMCI is not suitable for the analysis of dynamical spatiotemporal gridded systems, such as climatological data, without significant preprocessing and downscaling of the data. PCMCI requires unrealistic sample sizes to achieve acceptable performance on even modestly sized problems and suffers from a notable curse of dimensionality. This work suggests that, even under generous structural assumptions, significant additional algorithmic improvements are needed before causal discovery algorithms can be reliably applied to grid-level outputs of earth system models.

54 ENVIRONMENTAL SCIENCES↗

A hybrid porous model for full reactor core scale CFD investigation of a prismatic HTGR

Three-dimensional (3-D) Computational Fluid Dynamics (CFD) analysis of a whole nuclear reactor core is a tremendous challenge due to the large geometric volume and complex structures. Here, this research presents a hybrid porous (HP) model to simplify a prismatic High Temperature Gas-cooled Reactor (HTGR) core, so 3-D CFD investigation can be performed on a full reactor core scale. In the HP model, the prototypic small coolant channels in the nuclear fuel blocks are lumped together to form multiple equivalent large coolant channels, and then the porous medium flow model is applied to each of them. Therefore, heat transfer in fuel blocks is computed by a hybrid combination of solid energy and porous flow energy equations. The similarity between the HP model and prototypic model is achieved by deriving the porous flow permeability, inertial resistance factor, and artificial thermophysical properties. Compared with the widely used whole porous (WP) flow model, the HP model preserves more realistic geometric structures, and therefore more accurate physical processes. The General Atomics' Modular High Temperature Gas-cooled Reactor (MHTGR) design was chosen as a prototype to demonstrate the methodology. Simulations were performed using the prototypic CFD model and HP model at steady-state forced circulation, steady-state natural circulation, and transient conditions that correspond to normal operation, extended period of pressurized cool down, and short-term transients after reactor shutdown, respectively. The comparison shows good agreement between the HP model and prototypic model in the maximum fuel temperature, average solid temperature, and helium flow rate, which demonstrates the potential applicability of the HP model for a full reactor core scale simulation in the future. As a benefit, the HP model reduces the mesh quantity by a factor of 50 from a prototypic model. Correspondingly, the computation time was reduced by a factor of at least 30.

22 GENERAL STUDIES OF NUCLEAR REACTORS↗

PORFLOW modeling of Vadose zone flow and transport for the E-area intermediate level vault

In support of the E-Area Performance Assessment, a two-dimensional model of water flow and radionuclide transport through the E-Area Intermediate Level Vault (ILV) and local vadose zone has been developed using the PORFLOWTM software. The purpose of the model is to calculate flux to the water table for radionuclides eluted from the ILV during its operational life, the period of institutional control, and times following site closure. Results of model calculations will be used by a three-dimensional PORFLOW model of transport through the aquifer to determine radionuclide concentrations at a hypothetical 100 meter well and at the site boundary where contaminated groundwater could be assessed by members of the public following site closure. While newly developed, the model structure closely follows that used in the 2008 PA while incorporating a refined computational mesh, updated material properties for the vadose zone soil and vault concrete, revised infiltration rates, and new solid-liquid distribution coefficients. The model also addresses degradation of concrete hydraulic properties by blending soil and concrete water retention curves over a 500 year period. That is, the hydraulic properties of ILV concrete start out as fresh concrete at the time of site closure and degrade to soil properties over a 500 year time period. This approach has not been used previously in E-Area PA’s. The model has been used to calculate water flow through the ILV vadose zone and, on a trial basis, transport of a limited number of radionuclides from the waste region inside the ILV to the water table. Figure ES-1 provides a view of the model and an example of water flow after the concrete has substantially degraded. Figure dimensions are in centimeters. The red area in the left hand side figure is the waste disposal region within the ILV. Colored bands in the figures indicate different soil and concrete regions. Figure ES-2 shows a calculation of flux to the water table (mol/yr) for U-234 and its one-year half-life decay chain (U-234 → Th-230 → Ra-226 → Pb-210). Pb-210 flux is too small to show on the same scale used for Th-230 and Ra-226 in Figure ES-2. The model has also been used to make a preliminary evaluation of the release of tritium from TPBAR disposal containers that will be place in the ILV. A separately calculated tritium release rate as a function of time was used as a source in the PORFLOW model. The TPBAR disposal schedule and tritium release rate differ from the values used in the 2008 PA. Results from these initial trial runs are compared to calculations made for the 2008 PA ILV analysis. In brief, a new PORFLOW model of flow and transport through the ILV vadose zone has been developed and tested. The model is ready to be used for PA calculations.

12 MANAGEMENT OF RADIOACTIVE AND NON-RADIOACTIVE W↗

A Macroalgal Cultivation Modeling System (MACMODS): Evaluating the Role of Physical-Biological Coupling on Nutrients and Farm Yield

Offshore aquaculture has the potential to expand the macroalgal industry. However, moving into deeper waters requires suspended structures that will present novel farm-environment interactions. Here, we present a computational modeling framework, the Macroalgal Cultivation Modeling System (MACMODS), to explore within-farm modifications to light, seawater flow, and nutrient fields across time and space scales relevant to macroalgae. A regional ocean model informs the site-specific setting, the Santa Barbara Channel in the Southern California Bight. A fine-scale hydrodynamic model predicts modified flows and turbulent mixing within the farm. A spatially resolved macroalgal growth model, parameterized for giant kelp, Macrocystis pyrifera , predicts kelp biomass. Key findings from model integration are that regional ocean conditions set overall farm performance, while fine-scale within-farm circulation and nutrient delivery are important to resolve variation in within-farm macroalgal performance. Therefore, we conclude that models resolving within-farm dynamics can provide benefit to farmers with insight on how farm design and regional ocean conditions interact to influence overall yield. Here, the presence of repeating longlines aligned with the mean current generate flow diversions around the farm as well as attached Langmuir circulations and increased turbulence intensity. These flow-induced phenomena lead to less biomass in the interior portion of the farm relative to the edges. We also find that there is an effluent “footprint” that extends as much as 20 km beyond the farm. In this regard, MACMODS can be used to not only evaluate farm design and cultivation practices that maximize yield but also explore interactions between the farm and ecosystem in order to minimize impacts.

54 ENVIRONMENTAL SCIENCES↗