Networks Lines And Fields

[Semarias Alfna]

Iam pretty new to artificial intelligence and neural networks. I have implemented a feed-forward neural network in PyTorch for classification on the MNIST data set. Now I want to visualize the receptive fields of (a subset of) hidden neurons. But I am having some problems with understanding the concept of receptive fields and when I google it all results are about CNNs. So can anyone help me with how I could do this in PyTorch and how to interpret the results?

It seems that you are also struggling with the idea of receptive fields. Generally, you can best understand it by asking the question "which part of the (previous) layer representation is affecting my current input?"

In Convolutional layers, the formula to compute the current layer only takes part of the image as an input (or at least only changes the outcome based on a this subregion). This is precisely what the receptive field is.

Now, a fully connected layer, as the name implies, has a connection from every previous hidden state to every new hidden state, see the image below:

In that case, the receptive field is simply "every previous state" (e.g., in the image, a cell in the first turquoise layer is affected by all yellow cells), which is not very helpful. The whole idea would be to have a smaller subset instead of all available states.

Although we endeavor to make our web sites work with a wide variety of browsers, we can only support browsers that provide sufficiently modern support for web standards. Thus, this site requires the use of reasonably up-to-date versions of Google Chrome, FireFox, Internet Explorer (IE 9 or greater), or Safari (5 or greater). If you are experiencing trouble with the web site, please try one of these alternative browsers. If you need further assistance, you may write to he...@aps.org.

Our framework greatly extends the reach of tensor-network methods to study problems posed by continuous systems, allowing researchers to directly study various quantum field theories without the need for a prior coarse-graining of space. Reciprocally, it also allows researchers to import continuum techniques, hybridized with tensor networks, to approximately solve systems on a lattice.

Functional integral representation. In the discrete (left), a tensor network state is obtained from a contraction of auxiliary indices connecting the elementary tensors with each other and with a boundary tensor. In the continuum (right), the contraction is replaced by a functional integral (1), the auxiliary indices by fields ϕ, and the boundary tensor by a boundary functional B.

Tensor blocking. In d=1, blocking does not increase the bond dimension. In d=2, going from the UV to the IR doubles the bond dimension at each blocking. Hence, flowing the other way, from IR to UV, one reaches a trivial bond dimension after a finite number of iterations unless the initial bond dimension is infinite.

Gauge transformations. In the discrete case (left), transforming the elementary tensor as in Eq. (49) has a nontrivial result on the boundary only. In the continuum (right), the transformation of the elementary tensor is equivalent to the addition of a pure divergence term for the auxiliary fields in the bulk, which can then be integrated into a boundary condition.

Physical states on a boundary. In the discrete (left), the MERA is a (special case of a) tensor network state with a hierarchical structure, with physical indices only at the boundary. Tensor network states with such a structure can also be extended to the continuum with our CTNS ansatz, by restricting the physical field to the boundary and choosing an appropriate metric (here hyperbolic) for the bulk auxiliary fields.

It is not necessary to obtain permission to reuse thisarticle or its components as it is available under the terms ofthe Creative Commons Attribution 4.0 International license.This license permits unrestricted use, distribution, andreproduction in any medium, provided attribution to the author(s) andthe published article's title, journal citation, and DOI aremaintained. Please note that some figures may have been included withpermission from other third parties. It is your responsibility toobtain the proper permission from the rights holder directly forthese figures.

Nested sampling is a promising method for calculating phase diagrams of materials. However, if accuracy at the level of ab initio calculations is required, the computational cost limits its applicability. In the present work, we report on the efficient use of a neural-network force field in conjunction with the nested-sampling algorithm. We train our force fields on a recently reported database of silicon structures evaluated at the level of density functional theory and demonstrate our approach on the low-pressure region of the silicon pressure-temperature phase diagram between 0 and 16GPa. The simulated phase diagram shows good agreement with experimental results, closely reproducing the melting line. Furthermore, all of the experimentally stable structures within the investigated pressure range are also observed in our simulations. We point out the importance of the choice of exchange-correlation functional for the training data and show how the r2SCAN meta-generalized gradient approximation plays a pivotal role in achieving accurate thermodynamic behavior. We furthermore perform a detailed analysis of the potential energy surface exploration and highlight the critical role of a diverse and representative training data set.

Synthetic scenario illustrating the advantage of the dask parallelization scheme. Boxes delimited by thin lines indicate individual cell steps or GMC atom trajectories. Note that the larger boxes correspond to the more time-consuming atom move trajectories. Boxes delimited by thick lines indicate whole random walks of particular walkers. For better visibility, the latter are also colored.

The first two principal components of spherical Bessel descriptors highlighting different silicon phases present in the structural database. Each gray triangle represents one structure; colored areas show the convex hull around sets of structures corresponding to a certain phase. Only a few example phases are highlighted.

(a) Force and (b) energy training and validation set parity plots for the NNFF. Energies per atom are given relative to the minimum energy occurring in the respective dataset. (c) Energy-volume curves for several crystalline phases of silicon evaluated using DFT with a certain functional (solid lines) and corresponding NNFF models (dotted lines) that were trained on a database evaluated using the same functional. Top: r2SCAN. Bottom: PBE. Energies per atom are given relative to the minimum energy of the cubic diamond phase for both functionals.

Evolution of the walker population of the 32-atom, seed = 0, simulation at 10GPa in configuration space over time illustrated using the 2D configuration space map from Fig. 2. Gray points correspond to configurations in the training database; dashed lines indicate the convex hull of configurations belonging to a certain phase. Blue lines show the trajectory of NS samples (plotting only every 10 000th sample). Red points show the walker population at the given iteration. Histograms to the side show the population of space groups determined for optimized walkers.

Analysis of the basins that are explored during the nested sampling for the 32-atom, seed = 0, simulations. (a) Population of the most prominent space groups over time for the walkers at each simulated pressure. Colors indicate the iteration. (b) Partition function ratios of different occurring phases for all simulated pressure values, averaged over all three independent 32-atom runs, excluding two outliers discussed in the text. Colored areas show the standard deviation. Dashed lines show averaged heat capacity cp peak positions corresponding to melting (see Table 1).

Thank you for visiting nature.com. You are using a browser version with limited support for CSS. To obtain the best experience, we recommend you use a more up to date browser (or turn off compatibility mode in Internet Explorer). In the meantime, to ensure continued support, we are displaying the site without styles and JavaScript.

Transcriptional regulation is essential to life and is orchestrated by complex arrays of protein and RNA molecules. The most basic type of transcriptional regulation is exerted by transcription factor proteins that bind regulatory sequences of genes and affect their expression1. Elucidation of transcriptional regulatory systems is important for improving medicine and agriculture. For example, many diseases are associated with mutations in transcriptional regulators or in transcription factor binding sequences1. Changes in plant transcriptional regulation led to many modern crops and enabled large yield increases2. A better understanding of transcriptional regulation could help improve many agronomical traits such as biomass and resilience against pathogens3. Computational reverse engineering of gene regulatory networks has gained much attention over the last decade, driven by the emergence of large-scale gene expression analyses4,5. However, gene regulatory network inference remains a challenging task. This is in part due to the large amount of experimental noise and the large number of genes relative to the small sets of conditions in gene expression analyses6,7. In eukaryotes, gene expression levels are further affected by chromatin remodeling, and post-transcriptional and post-translational processes8. All these additional layers of regulation make inference of causal dependencies between genes from gene expression datasets alone even more difficult. While inference methods for in silico and prokaryotic datasets perform well5, inferring gene regulatory networks from eukaryotic datasets is more difficult5,9,10. As a consequence, heterogeneous data integration methods11 have emerged to construct more reliable eukaryotic biological networks for gene function prediction11,12,13 and gene regulatory network inference14.

3a8082e126