On the Hyperparameters influencing a PINN's generalization beyond the training domain

Physics-Informed Neural Networks (PINNs) are Neural Network architectures trained to emulate solutions of differential equations without the necessity of solution data. They are currently ubiquitous in the scientific literature due to their flexible and promising settings. However, very little of the available research provides practical studies that aim for a better quantitative understanding of such architecture and its functioning. In this paper, we analyze the performance of PINNs for various architectural hyperparameters and algorithmic settings based on a novel error metric and other factors such as training time. The proposed metric and approach are tailored to evaluate how well a PINN generalizes to points outside its training domain. Besides, we investigate the effect of the algorithmic setup on the outcome prediction of a PINN, inside and outside its training domain, to explore the effect of each hyperparameter. Through our study, we assess how the algorithmic setup of PINNs influences their potential for generalization and deduce the settings which maximize the potential of a PINN for accurate generalization. The study that we present returns insightful and at times counterintuitive results on PINNs. These results can be useful in PINN applications when defining the model and evaluating it

Measuring Accessibility to Medical Centers in Isfahan City Using 2SFCA Method

AbstractOne of the most important challenges facing policymakers and urban planners in recent decades is the issue of accessibility to a variety of urban services. The main purpose of this study was thecalculation of the accessibility of census blocks to medical centers using the Two-Step Floating Catchment Area (2SFCA) method in Isfahan City. In the present study, according to the conditions with and without the limitations of the accessibility radii, different types of distance decay functions were used. The results showed that the 2SFCA method with the use of the cumulative opportunity negative linear function had the highest average of correlation for calculating accessibility to medical centers in comparison with other functions. Calculation of average accessibility in the 15 main regions of Isfahan City showed that the central regions (3, 1, and 5) had the highest decrease and the marginal regions (9, 8, and 11) had the highest increase in the unlimited compared to the limited mode. In general, based on the obtained results of 2SFCA method and the calculated Gini index, the level of inequality in accessibility of census blocks to health services was high in Isfahan City and this inequality increased in terms of accessibility to both hospitals and clinics. Since the extended 2SFCA method has a high capability for assessing supply and demand, as well as catchment area, application of this method can provide a great help for managers and planners in theassessment of the population’s access to a variety of services, such as emergency services and healthcare.Keywords: spatial accessibility, 2SFCA method, distance decay function, medical centers, Isfahan IntroductionOne of the most important challenges faced by policymakers and urban planners in recent decades has been the subjct of access to a variety of urban services. Hospital and clinic centers as the most important urban facilities play an important role in serving people. handeling access to healthcare requires examining the factors, such as spatial distribution of services and demands. Distribution of healthcare centers can affect ease of accessibility for applicants. As health is the basis of social, economic, political, and cultural developments of human societies, identifying deprived areas in terms of accessibility and planning for equitable accessibility to health services for all members of society are essential. MethodologyIn the present study, the Two-Step Floating Catchment Area Method (2SFCA) was employed to calculate the access of census blocks to medical centers (hospitals and clinics) in the city of Isfahan for limited and unlimited accessibility radii. To define the most appropriate distance decay function in the 2SFCA method, the average of Pearson’s correlation coefficient between the accessibility values ​​obtained from different distance decay functions was used. The distance decay function with the highest mean correlation of accessibility values compared to other functions was determined as the most appropriate function in the 2SFCA method. Also, the Lorenz curve and Gini coefficient were applied to compare inequalities of access to medical centers in Isfahan. Results and DiscussionThe results showed that the use of the negative linear cumulative opportunity distance decay function had the highest average correlation in the accessibility values compared to other functions. In the case of limited accessibility radius, the central regions and some northwest and east areas had the highest accessibility to hospitals. In the case of unlimited radius, the central areas had the most accessibility, while accessibility decreased as the distance from these areas increased. Calculation of the average accessibility in the 15 main regions of Isfahan showed that the central (3, 1 and 5) and marginal (9, 8, and 11) regions had the highest decrease and increase in the unlimited compared to the limited mode, respectively. Also, the sensitivity analysis of accessibility to hospitals showed that Al-Zahra and Hazrat Zahra hospitals in Districts 5 and 14 had the greatest impacts on the accessibility of cesus blocks to hospital services in Isfahan City. Comparing the accessibility of census blocks to both hospitals and clinics with accessibility only to hospitals showed an increase in accessibility in the central areas of the city due to the greater concentration of clinics in those areas. However, in the case of combination of hospitals and clinics, the Gini coefficient was equal to 0.60, which showed an increase of 0.04 compared to the case of accessibility only to hospitals, which indicated that inequality was higher in the combinatorial case. ConclusionConsidering the supply and demand simultaneously, the 2SFCA method can provide a more realistic assessment of the accessibility status of census blocks to medical services. In general, based on the obtained results by this method and due to considering the limited radius of accessibility and calculating the Gini index, the level of inequality in the accessibility of census blocks to health services was high in Isfahan City, while this inequality increased in the case of  accessibility to both hospitals and clinics. References- Apparicio, P., Gelb, J., Dubé, A. S., Kingham, S., Gauvin, L., & Robitaille, É. (2017). The approaches to measuring the potential spatial access to urban health services revisited: distance types and aggregation-error issues. International Journal of Health Geographics, 16(1), 1-24.- Bryant Jr, J. and Delamater, P. L. (2019). Examination of spatial accessibility at micro- and macro-levels using the enhanced two-step floating catchment area (E2SFCA) method. Annals of GIS, 25(3), 219-229.- Chatterjee, S. and Hadi, A. S. (2006). Regression analysis by example. 4th Ed., John Wiley & Sons.- Chen, X. and Jia, P. (2019). 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Relationship between thyroid hormone levels in euthyroid patients before HSCT and time to achieve neutrophil and platelet engraftment: an analytical cross-sectional study

Introduction. The time to reach neutrophil (NE) and platelet engraftment (PE) in hematopoietic stem cell transplantation (HSCT) is one of the most important factors indicating transplantation survival. The aim of this study was to investigate the relationship between thyroid hormone levels before HSCT and the time to achieve NE and PE. Material and methods. The relationship between thyroid hormone levels before HSCT, age, gender, type of HSCT, type of disease and cluster of differentiation 34+ (CD34+) cell count and the number of days to reach NE and PE was examined in 37 clinically and laboratorially euthyroid patients. Results. An odds ratio (OR) of > 6 was observed in the probability of time to NE > 10 days in patients with thyroid-stimulating hormone (TSH) > 2.89 mU/L in the upper normal range (UNR) and male patients, also in the probability of time to PE > 15 days in patients with TSH > 2.89 mU/L in the UNR. Statistically significant p-value and confidence interval were found in the probability of time to NE > 10 days in male patients (OR = 8.58, p-value = 0.036) and time to PE > 15 days in patients with TSH > 2.89 mU/L in the UNR (OR = 14.32, p-value = 0.041). Conclusions. Treatment with low dose levothyroxine can be cautiously recommended to achieve TSH to ≤2.8 mU/L in the lower normal range before performing HSCT in euthyroid patients, which will reduce the times to NE and PE and help earlier discharge of patients

Search for gravitational wave bursts in LIGO's third science run

We report on a search for gravitational wave bursts in data from the three LIGO interferometric detectors during their third science run. The search targets subsecond bursts in the frequency range 100-1100 Hz for which no waveform model is assumed, and has a sensitivity in terms of the root-sum-square (rss) strain amplitude of hrss ~ 10^{-20} / sqrt(Hz). No gravitational wave signals were detected in the 8 days of analyzed data.Comment: 12 pages, 6 figures. Amaldi-6 conference proceedings to be published in Classical and Quantum Gravit

All-sky search for periodic gravitational waves in LIGO S4 data

We report on an all-sky search with the LIGO detectors for periodic gravitational waves in the frequency range 50-1000 Hz and with the frequency's time derivative in the range -1.0E-8 Hz/s to zero. Data from the fourth LIGO science run (S4) have been used in this search. Three different semi-coherent methods of transforming and summing strain power from Short Fourier Transforms (SFTs) of the calibrated data have been used. The first, known as "StackSlide", averages normalized power from each SFT. A "weighted Hough" scheme is also developed and used, and which also allows for a multi-interferometer search. The third method, known as "PowerFlux", is a variant of the StackSlide method in which the power is weighted before summing. In both the weighted Hough and PowerFlux methods, the weights are chosen according to the noise and detector antenna-pattern to maximize the signal-to-noise ratio. The respective advantages and disadvantages of these methods are discussed. Observing no evidence of periodic gravitational radiation, we report upper limits; we interpret these as limits on this radiation from isolated rotating neutron stars. The best population-based upper limit with 95% confidence on the gravitational-wave strain amplitude, found for simulated sources distributed isotropically across the sky and with isotropically distributed spin-axes, is 4.28E-24 (near 140 Hz). Strict upper limits are also obtained for small patches on the sky for best-case and worst-case inclinations of the spin axes.Comment: 39 pages, 41 figures An error was found in the computation of the C parameter defined in equation 44 which led to its overestimate by 2^(1/4). The correct values for the multi-interferometer, H1 and L1 analyses are 9.2, 9.7, and 9.3, respectively. Figure 32 has been updated accordingly. None of the upper limits presented in the paper were affecte

All-sky LIGO Search for Periodic Gravitational Waves in the Early S5 Data

We report on an all-sky search with the LIGO detectors for periodic gravitational waves in the frequency range 50--1100 Hz and with the frequency's time derivative in the range -5.0E-9 Hz/s to zero. Data from the first eight months of the fifth LIGO science run (S5) have been used in this search, which is based on a semi-coherent method (PowerFlux) of summing strain power. Observing no evidence of periodic gravitational radiation, we report 95% confidence-level upper limits on radiation emitted by any unknown isolated rotating neutron stars within the search range. Strain limits below 1.E-24 are obtained over a 200-Hz band, and the sensitivity improvement over previous searches increases the spatial volume sampled by an average factor of about 100 over the entire search band. For a neutron star with nominal equatorial ellipticity of 1.0E-6, the search is sensitive to distances as great as 500 pc--a range that could encompass many undiscovered neutron stars, albeit only a tiny fraction of which would likely be rotating fast enough to be accessible to LIGO. This ellipticity is at the upper range thought to be sustainable by conventional neutron stars and well below the maximum sustainable by a strange quark star.Comment: 6 pages, 1 figur

Search for gravitational waves from binary inspirals in S3 and S4 LIGO data

We report on a search for gravitational waves from the coalescence of compact binaries during the third and fourth LIGO science runs. The search focused on gravitational waves generated during the inspiral phase of the binary evolution. In our analysis, we considered three categories of compact binary systems, ordered by mass: (i) primordial black hole binaries with masses in the range 0.35 M(sun) < m1, m2 < 1.0 M(sun), (ii) binary neutron stars with masses in the range 1.0 M(sun) < m1, m2 < 3.0 M(sun), and (iii) binary black holes with masses in the range 3.0 M(sun)< m1, m2 < m_(max) with the additional constraint m1+ m2 < m_(max), where m_(max) was set to 40.0 M(sun) and 80.0 M(sun) in the third and fourth science runs, respectively. Although the detectors could probe to distances as far as tens of Mpc, no gravitational-wave signals were identified in the 1364 hours of data we analyzed. Assuming a binary population with a Gaussian distribution around 0.75-0.75 M(sun), 1.4-1.4 M(sun), and 5.0-5.0 M(sun), we derived 90%-confidence upper limit rates of 4.9 yr^(-1) L10^(-1) for primordial black hole binaries, 1.2 yr^(-1) L10^(-1) for binary neutron stars, and 0.5 yr^(-1) L10^(-1) for stellar mass binary black holes, where L10 is 10^(10) times the blue light luminosity of the Sun.Comment: 12 pages, 11 figure

Search for Gravitational Wave Bursts from Soft Gamma Repeaters

We present the results of a LIGO search for short-duration gravitational waves (GWs) associated with Soft Gamma Repeater (SGR) bursts. This is the first search sensitive to neutron star f-modes, usually considered the most efficient GW emitting modes. We find no evidence of GWs associated with any SGR burst in a sample consisting of the 27 Dec. 2004 giant flare from SGR 1806-20 and 190 lesser events from SGR 1806-20 and SGR 1900+14 which occurred during the first year of LIGO's fifth science run. GW strain upper limits and model-dependent GW emission energy upper limits are estimated for individual bursts using a variety of simulated waveforms. The unprecedented sensitivity of the detectors allows us to set the most stringent limits on transient GW amplitudes published to date. We find upper limit estimates on the model-dependent isotropic GW emission energies (at a nominal distance of 10 kpc) between 3x10^45 and 9x10^52 erg depending on waveform type, detector antenna factors and noise characteristics at the time of the burst. These upper limits are within the theoretically predicted range of some SGR models.Comment: 6 pages, 1 Postscript figur

Quantum state preparation and macroscopic entanglement in gravitational-wave detectors

Long-baseline laser-interferometer gravitational-wave detectors are operating at a factor of 10 (in amplitude) above the standard quantum limit (SQL) within a broad frequency band. Such a low classical noise budget has already allowed the creation of a controlled 2.7 kg macroscopic oscillator with an effective eigenfrequency of 150 Hz and an occupation number of 200. This result, along with the prospect for further improvements, heralds the new possibility of experimentally probing macroscopic quantum mechanics (MQM) - quantum mechanical behavior of objects in the realm of everyday experience - using gravitational-wave detectors. In this paper, we provide the mathematical foundation for the first step of a MQM experiment: the preparation of a macroscopic test mass into a nearly minimum-Heisenberg-limited Gaussian quantum state, which is possible if the interferometer's classical noise beats the SQL in a broad frequency band. Our formalism, based on Wiener filtering, allows a straightforward conversion from the classical noise budget of a laser interferometer, in terms of noise spectra, into the strategy for quantum state preparation, and the quality of the prepared state. Using this formalism, we consider how Gaussian entanglement can be built among two macroscopic test masses, and the performance of the planned Advanced LIGO interferometers in quantum-state preparation

Searching for a Stochastic Background of Gravitational Waves with LIGO

The Laser Interferometer Gravitational-wave Observatory (LIGO) has performed the fourth science run, S4, with significantly improved interferometer sensitivities with respect to previous runs. Using data acquired during this science run, we place a limit on the amplitude of a stochastic background of gravitational waves. For a frequency independent spectrum, the new limit is $\Omega_{\rm GW} < 6.5 \times 10^{-5}$. This is currently the most sensitive result in the frequency range 51-150 Hz, with a factor of 13 improvement over the previous LIGO result. We discuss complementarity of the new result with other constraints on a stochastic background of gravitational waves, and we investigate implications of the new result for different models of this background.Comment: 37 pages, 16 figure