We present GravLensX, an innovative method for rendering black holes with gravitational lensing effects using neural networks. The methodology involves training neural networks to fit the spacetime around black holes and then employing these trained models to generate the path of light rays affected by gravitational lensing. This enables efficient and scalable simulations of black holes with optically thin accretion disks, significantly decreasing the time required for rendering compared to traditional methods. We validate our approach through extensive rendering of multiple black hole systems with superposed Kerr metric, demonstrating its capability to produce accurate visualizations with significantly $15\times$ reduced computational time. Our findings suggest that neural networks offer a promising alternative for rendering complex astrophysical phenomena, potentially paving a new path to astronomical visualization.

The transform is fundamental tool in science, but also is how your eyes see the world. There is a great YouTube channel called 3Blue1Brown. I would bet that you've heard of it, but in case not, it is a very popular math education channel by Grant Sanderson that heavily uses visualizations, with a particular style that has noticeably influenced many other YouTubers. I bring this up at the beginning because he recently did a set of videos on the Fourier Transform (FT), with a particular focus on explaining the Fourier-convolution theorem, which is pretty relevant to topics I work on so it caught my interest. Like all of Sanderson's content, it is wonderfully put together, and pertinently it inspired me to finally write up this article about a Fourier-related topic.

"Oh, that looks gorgeous," drawls an onlooker in a Texas accent, as I guide my fighter through a gloomy, broken-down shack in a video game called Metro Exodus. I'm struggling in hand-to-hand combat with a zombie, so Dmitry Zhdan, who worked on the game, grabs the controls and beats my foe to death with a rifle butt.

Modern films and TV shows are filled with spectacular computer-generated sequences computed by rendering systems that simulate the flow of light in a three-dimensional scene and convert the information into a two-dimensional image. But computing the thousands of light rays (per frame) to achieve accurate colour, shadows, reflectivity and other light-based characteristics is a labour-intensive, time-consuming and expensive undertaking. An alternative is to render the images using only a few light rays. That saves time and labour but results in inaccuracies that show up as objectionable "noise" in the final image. UC Santa Barbara electrical and computer engineering Ph.D. student Steve Bako and his advisor, Pradeep Sen, are advancing on a solution.

Google uses "deep learning" to generate captions for images, Facebook uses it to recognize faces and Tesla uses it to train self-driving cars. Now astronomers have caught on to deep learning, a form of machine learning in which a computer can be trained to identify or classify particular objects in images. The newest telescopes, such as the Dark Energy Survey, which uses a 4-meter telescope in northern Chile and covers about one quarter of the southern sky, take millions of images of a variety of celestial objects. These often include visual distortions, cosmic rays and satellite trails that make them difficult to interpret. Deep learning could help process this deluge of data quickly.

The modest eyeless roundworm appears to have the remarkable superhero-like ability to taste light. A new type of cell has been discovered in the taste receptors of these creatures which is about 50 times more efficient at capturing light than the human eye. Scientists say that the unusual characteristics of the receptor could be used in a range of applications, including creating more effective sunscreens. It might sound like a power only superheroes could have, but the modest eyeless roundworm has the ability to taste light. Animal photoreceptors usually have two components – a base protein, and a light-absorbing chromophore.

In Einstein's general theory of relativity, there's no conceptual distinction between the past and the future, let alone an objective line of "now." There's also no sense in which time "flows"; instead, all of space and time is just there in some four-dimensional structure. None of these facts are easy to accept, because they're in direct conflict with our subjective experience of time. But don't feel too bad: They're hard even for physicists to accept, an ongoing tension that places physics in conflict not just with common sense but also with itself. As much as physicists talk about time symmetry, they do not allow themselves to invoke the future, only the past, when seeking to explain occurrences in the world.

We propose a special computational device which uses light rays for solving the subset-sum problem. The device has a graph-like representation and the light is traversing it by following the routes given by the connections between nodes. The nodes are connected by arcs in a special way which lets us to generate all possible subsets of the given set. To each arc we assign either a number from the given set or a predefined constant. When the light is passing through an arc it is delayed by the amount of time indicated by the number placed in that arc. At the destination node we will check if there is a ray whose total delay is equal to the target value of the subset sum problem (plus some constants).

The sum-product algorithm can be directly applied in Gaussian networks and in graphs for coding, but for many conditional probabilityfunctions - including the sigmoid function - direct application of the sum-product algorithm is not possible. We introduce "accumulator networks" that have low local complexity (but exponential global complexity) so the sum-product algorithm can be directly applied. In an accumulator network, the probability of a child given its parents is computed by accumulating the inputs from the parents in a Markov chain or more generally a tree. After giving expressions for inference and learning in accumulator networks, wegive results on the "bars problem" and on the problem of extracting translated, overlapping faces from an image. 1 Introduction Graphical probability models with hidden variables are capable of representing complex dependenciesbetween variables, filling in missing data and making Bayesoptimal decisionsusing probabilistic inferences (Hinton and Sejnowski 1986; Pearl 1988; Neal 1992). Large, richly-connected networks with many cycles can potentially beused to model complex sources of data, such as audio signals, images and video. However, when the number of cycles in the network is large (more precisely, when the cut set size is exponential), exact inference becomes intractable. Also, to learn a probability model with hidden variables, we need to fill in the missing data using probabilistic inference, so learning also becomes intractable. To cope with the intractability of exact inference, a variety of approximate inference methods have been invented, including Monte Carlo (Hinton and Sejnowski 1986; Neal 1992), Helmholz machines (Dayan et al. 1995; Hinton et al. 1995), and variational techniques (Jordan et al. 1998).