Differentiable programming for scientific machine learning (SciML) has recently seen considerable interest and success, as it directly embeds neural networks inside PDEs, often called as NeuralPDEs, derived from first...
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Differentiable programming for scientific machine learning (SciML) has recently seen considerable interest and success, as it directly embeds neural networks inside PDEs, often called as NeuralPDEs, derived from first principle physics. Unlike large, parameterized black-box deep neural networks, NeuralPDEs promise a targeted and efficient learning approach by only representing unknown terms and allowing the rest of the known PDE to constrain the network with known physics. Owing to these strengths, there is a widespread assumption in the community that NeuralPDEs are more trustworthy and generalizable than black box models. However, like any SciML model, differentiable programming relies predominantly on high-quality PDE simulations as "ground truth" for training. In the era of foundation models, the blanket assumption of ground truth has been made for any scientific simulation that satisfactorily captures physical phenomena. However, mathematics dictates that these are only discrete numerical approximations of the true physics in nature. Therefore, we must pose the following questions: Are NeuralPDEs and differentiable programming models trained on PDE simulations indeed as physically interpretable as we think? And in cases where the NeuralPDEs can successfully extrapolate, are they doing so for the right reasons? In this work, we rigorously attempt to answer these questions, using established ideas from numerical analysis and dynamical systems theory. We use (1+1)-dimensional PDEs as our test cases: the viscous Burgers equation, and the geophysical Kortveg de Vries equations. Our analysis shows that NeuralPDEs learn the artifacts in the simulation training data arising from the discretized Taylor Series truncation error of the spatial derivatives. Consequently, we find that NeuralPDE models are systematically biased and their generalization capability likely results from, instead of learning physically relevant quantities, a fortuitous interplay of numerical dissi
In this paper, we prove Strichartz estimates for many body Schrödinger equations in the periodic setting, specifically on tori Td, where d ≥ 3. The results hold for both rational and irrational tori, and for sma...
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In this paper we consider the modeling of measurement error for fund returns data. In particular, given access to a time-series of discretely observed log-returns and the associated maximum over the observation period...
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Pasteurellosis remains a major problem for poultry worldwide. The disease causes high mortality in chicken, affecting the livelihood of rural poultry farmers. In this paper, both deterministic and continuous time Mark...
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Disordered hyperuniform many-particle systems are recently discovered exotic states of matter, characterized by a complete suppression of normalized infinite-wavelength density fluctuations, as in perfect crystals, an...
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Disordered hyperuniform many-particle systems are recently discovered exotic states of matter, characterized by a complete suppression of normalized infinite-wavelength density fluctuations, as in perfect crystals, and lack of conventional long-range order, as in liquids and glasses. In this work, we begin a program to quantify the structural properties of nonhyperuniform and hyperuniform networks. In particular, large two-dimensional (2D) Voronoi networks (graphs) containing approximately 10,000 nodes are created from a variety of different point configurations, including the antihyperuniform hyperplane intersection process (HIP), nonhyperuniform Poisson process, nonhyperuniform random sequential addition (RSA) saturated packing, and both non-stealthy and stealthy hyperuniform point processes. We carry out an extensive study of the Voronoi-cell area distribution of each of the networks through determining multiple metrics that characterize the distribution, including their higher-cumulants (i.e., skewness 1 and excess kurtosis 2). We show that the HIP distribution is far from Gaussian, as evidenced by a high skewness (γ1 = 3.16) and large positive excess kurtosis (γ2 = 16.2). The Poisson (with γ1 = 1.07 and γ2 = 1.79) and non-stealthy hyperuniform (with γ1 = 0.257 and γ2 = 0.0217) distributions are Gaussian-like distributions, since they exhibit a small but positive skewness and excess kurtosis. The RSA (with γ1 = 0.450 and γ2 = -0.0384) and the highest stealthy hyperuniform distributions (with γ1 = 0.0272 and γ2 = -0.0626) are also non-Gaussian because of their low skewness and negative excess kurtosis, which is diametrically opposite non-Gaussian behavior of the HIP. The fact that the cell-area distributions of large, finite-sized RSA and stealthy hyperuniform networks (e.g., with N ≈ 10, 000 nodes) are narrower, have larger peaks, and smaller tails than a Gaussian distribution implies that in the thermodynamic limit the distributions should exhibit compact suppo
Industrial dyes are considered one of the main causes of increased water pollution of water. Many businesses, such as steel and paper, are located along riverbanks because they require large amounts of water in their ...
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Industrial dyes are considered one of the main causes of increased water pollution of water. Many businesses, such as steel and paper, are located along riverbanks because they require large amounts of water in their manufacturing processes, and their wastes, which contain acids, alkalis, dyes, and other chemicals, are dumped and poured into rivers as effluents. For example, chemical enterprises producing aluminum emit a significant quantity of fluoride into the air and effluents into water bodies. Fertilizer facilities produce a lot of ammonia, whereas steel plants produce cyanide. Many nations consider employing wastewater treatment plants using physical, biological, and chemical methods to clean the wastewater to address environmental crises. The treated water can be used for targeting the irrigation systems in its majority, as it is biologically acceptable for that specific use, industrial dyes are considered one of the leading causes of increased water pollution of water. Many businesses, such as steel and paper, are located along riverbanks because they require large amounts of water in their manufacturing processes, and their wastes, which contain acids, alkalis, dyes, and other chemicals, are dumped and poured into rivers as effluents. For example, chemical enterprises producing aluminum emit a significant quantity of fluoride into the air and effluents into water bodies. Fertilizer facilities produce much ammonia, whereas steel plants produce cyanide. Chromium salts are used in. Many nations consider employing wastewater treatment plants using physical, biological, and chemical methods to clean the wastewater to address environmental crises. The treated water can target the majority of irrigation systems, as it is biologically acceptable for that specific use, which economizes the use of freshwater sources for municipal use. This study presents a novel method for fabricating an efficient adsorbent sheet for wastewater treatment. The sheets are fabricated by
Understanding, predicting and controlling laminar-turbulent boundary-layer transition is crucial for the next generation aircraft design. However, in real flight experiments, or wind tunnel tests, often only sparse se...
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Understanding, predicting and controlling laminar-turbulent boundary-layer transition is crucial for the next generation aircraft design. However, in real flight experiments, or wind tunnel tests, often only sparse sensor measurements can be collected at fixed locations. Thus, in developing reduced models for predicting and controlling the flow at the sensor locations, the main challenge is in accounting for how the surrounding field of unobserved (or unresolved) variables interacts with the observed (or resolved) variables at the fixed sensor locations. This makes the Mori-Zwanzig (MZ) formalism a natural choice, as it results in the Generalized Langevin Equations which provides a mathematically sound framework for constructing non-Markovian reduced-order models that include the effects the unresolved variables have on the resolved variables. These effects are captured in the so called memory kernel and orthogonal dynamics, which, when using Mori's linear projection, provides a higher order approximation to the traditional approximate Koopman learning methods. In this work, we explore recently developed data-driven methods for extracting the MZ operators to two boundary-layer flows obtained from high resolution data;a low speed incompressible flow over a flat plate exhibiting bypass transition;and a high speed compressible flow over a flared cone at Mach 6 and zero angle of attack where transition was initiated using a broadband forcing approach ("natural" transition). In each case, an array of "sensors" are placed near the surface of the solid boundary, and the MZ operators are learned and the predictions are compared to the Extended Dynamic Mode Decomposition (EDMD), both using delay embedded coordinates. Further comparisons are made with Long Short-Term Memory (LSTM) and a regression based projection framework using neural networks for the MZ operators. First, we compare the effects of including delay embedded coordinates with EDMD and Mori based MZ and provide
BCC transition metals (TMs) exhibit complex temperature and strain-rate dependent plastic deformation behavior controlled by individual crystal lattice defects. Classical empirical and semiempirical interatomic potent...
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BCC transition metals (TMs) exhibit complex temperature and strain-rate dependent plastic deformation behavior controlled by individual crystal lattice defects. Classical empirical and semiempirical interatomic potentials have limited capability in modeling defect properties such as the screw dislocation core structures and Peierls barriers in the BCC structure. Machine learning (ML) potentials, trained on DFT-based datasets, have shown some successes in reproducing dislocation core properties. However, in group VB TMs, the most widely used DFT functionals produce erroneous shear moduli C44 which are undesirably transferred to machine-learning interatomic potentials, leaving current ML approaches unsuitable for this important class of metals and alloys. Here, we develop two interatomic potentials for BCC vanadium (V) based on (i) an extension of the partial electron density and screening parameter in the classical semiempirical modified embedded-atom method (XMEAM-V) and (ii) a recent hybrid descriptor in the ML Deep Potential framework (DP-HYB-V). We describe distinct features in these two disparate approaches, including their dataset generation, training procedure, weakness and strength in modeling lattice and defect properties in BCC V. Both XMEAM-V and DP-HYB-V reproduce a broad range of defect properties (vacancy, self-interstitials, surface, dislocation) relevant to plastic deformation and fracture. In particular, XMEAM-V reproduces nearly all mechanical and thermodynamic properties at DFT accuracies and with C44 near the experimental value. XMEAM-V also naturally exhibits the anomalous slip at 77 K widely observed in group VB and VIB TMs and outperforms all existing, publically available interatomic potentials for V. The XMEAM thus provides a practical path to developing accurate and efficient interatomic potentials for nonmagnetic BCC TMs and possibly multiprincipal element TM alloys.
作者：
Torquato, SalvatoreDepartment of Chemistry
Department of Physics Princeton Institute for the Science and Technology of Materials Program in Applied and Computational Mathematics Princeton University PrincetonNJ08544 United States
The study of hyperuniform states of matter is an emerging multidisciplinary field, impinging on topics in the physical sciences, mathematics and biology. The focus of this work is the exploration of quantitative descr...
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The study of hyperuniform states of matter is an emerging multidisciplinary field, impinging on topics in the physical sciences, mathematics and biology. The focus of this work is the exploration of quantitative descriptors that herald when a many-particle system in d-dimensional Euclidean space Rd approaches a hyperuniform state as a function of the relevant control parameter. We establish quantitative criteria to ascertain the extent of hyperuniform and nonhyperuniform distance-scaling regimes as well as the crossover point between them in terms of the "volume" coefficient A and "surface-area" coefficient B associated with the local number variance σ2(R) for a spherical window of radius R. The larger the ratio B/A, the larger the hyperuniform scaling regime, which becomes of infinite extent in the limit B/A → ∞. To complement the known direct-space representation of the coefficient B in terms of the total correlation function h(r), we derive its corresponding Fourier representation in terms of the structure factor S(k), which is especially useful when scattering information is available experimentally or theoretically. We also demonstrate that the free-volume theory of the pressure of equilibrium packings of identical hard spheres that approach a strictly jammed state either along the stable crystal or metastable disordered branch dictates that such end states be exactly hyperuniform. Using the ratio B/A, as well as other diagnostic measures of hyperuniformity, including the hyperuniformity index H and the direct-correlation function length scale ξc, we study three different exactly solvable models as a function of the relevant control parameter, either density or temperature, with end states that are perfectly hyperuniform. Specifically, we analyze equilibrium systems of hard rods and "sticky" hard-sphere systems in arbitrary space dimension d as a function of density. We also examine low-temperature excited states of many-particle systems interacting with "steal
In this paper, we establish an 2 decoupling inequality for the convex hypersurface{ (ξ1, ..., ξn−1, ξ1m + ... + ξmn−1): (ξ1, ..., ξn−1) ∈ [0, 1]n−1} associated with the decomposition adapted to hypersurfaces of...
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