Scientific Machine Learning (SciML) is a recently emerged research field which combines physics-based and data-driven models for the numerical approximation of differential problems. Physics-based models rely on the physical understanding of the problem at hand, subsequent mathematical formulation, and numerical approximation. Data-driven models instead aim to extract relations between input and output data without arguing any causality principle underlining the available data distribution. In recent years, data-driven models have been rapidly developed and popularized. Such a diffusion has been triggered by a huge availability of data (the so-called big data), an increasingly cheap computing power, and the development of powerful machine learning algorithms. SciML leverages the physical awareness of physics-based models and, at the same time, the efficiency of data-driven algorithms. With SciML, we can inject physics and mathematical knowledge into machine learning algorithms. Yet, we can rely on data-driven algorithms' capability to discover complex and non-linear patterns from data and improve the descriptive capacity of physics-based models. After recalling the mathematical foundations of digital modelling and machine learning algorithms, and presenting the most popular machine learning architectures, we discuss the great potential of a broad variety of SciML strategies in solving complex problems governed by partial differential equations. Finally, we illustrate the successful application of SciML to the simulation of the human cardiac function, a field of significant socio-economic importance that poses numerous challenges on both the mathematical and computational fronts. The corresponding mathematical model is a complex system of non-linear ordinary and partial differential equations describing the electromechanics, valve dynamics, blood circulation, perfusion in the coronary tree, and torso potential. Despite the robustness and accuracy of physics-based models, certain aspects, such as unveiling constitutive laws for cardiac cells and myocardial material properties, as well as devising efficient reduced order models to dominate the extraordinary computational complexity, have been successfully tackled by leveraging data-driven models.

To date, this advantage of RNNs over CNNs is entirely theoretical.

We analyze algorithmic and computational aspects of biological phenomena, such as replication and programmed death, in the context of machine learning. We use two different measures of neuron efficiency to develop machine learning algorithms for adding neurons to the system (i.e. replication algorithm) and removing neurons from the system (i.e. programmed death algorithm). We argue that the programmed death algorithm can be used for compression of neural networks and the replication algorithm can be used for improving performance of the already trained neural networks. We also show that a combined algorithm of programmed death and replication can improve the learning efficiency of arbitrary machine learning systems. The computational advantages of the bio-inspired algorithms are demonstrated by training feedforward neural networks on the MNIST dataset of handwritten images.

America is locked in a quantum computer race with China. The latest developments from Chinese scientists show a "significant computing breakthrough, achieving quantum computational advantage," according to state media Xinhua News Agency. Thursday, China's top quantum research group published a new research paper in the journal Science, titled "Quantum computational advantage using photons," outlines how a quantum computer prototype detected up to 76 photons through Gaussian boson sampling (GBS), a standard simulation algorithm, Xinhua said, adding that its ability to process complex problems is exponentially faster than most supercomputers. Called "Jiuzhang," the supercomputer prototype can conduct large-scale GBS 100 trillion times faster than the world's fastest supercomputer. Researchers said their prototype processes 10 billion times faster than the 53-qubit quantum computer developed by Google.

This similarity can be as simple as a categorical feature value such as the color or shape of the objects we are classifying, or a more complex function of all categorical and/or continuous feature values that these objects possess. Documents can be classified as well using their quantifiable attributes such as size, file extension etc… Easy! But unfortunately it is the meaning/import of the text contained in the document is what we are usually interested in for classification. The ingredients of text are words (and throw in punctuation as well) and the meaning of a text snippet is not a deterministic function of these constituents. We know that the same set of words but in a different order, or simply with different punctuation can convey different meanings.

A number of writers have supposed that for the full specification of belief, higher order probabilities are required. Some have even supposed that there may be an unending sequence of higher order probabilities of probabilities of probabilities.... In the present paper we show that higher order probabilities can always be replaced by the marginal distributions of joint probability distributions. We consider both the case in which higher order probabilities are of the same sort as lower order probabilities and that in which higher order probabilities are distinct in character, as when lower order probabilities are construed as frequencies and higher order probabilities are construed as subjective degrees of belief. In neither case do higher order probabilities appear to offer any advantages, either conceptually or computationally.

Machine intelligence problems are sometimes defined as those problems which (i) computers can't yet do, and (ii) humans can. We shall further consider how much "knowledge" about a finite mathematical function can, on certain assumptions, be credited to a computer program. Although our approach is quite general, we are really only interested in programs which evaluate "semihard" functions, believing that the evaluation of such functions constitutes the defining aspiration of machine intelligence work. If a function is less hard than "semihard," then we can evaluate it by pure algorithm (trading space for time) or by pure lookup (making the opposite trade), with no need to talk of knowledge, advice, machine intelligence, or any of those things. We call such problems "standard." If however the function is "semihard," then we will be driven to construct some form of artful compromise between the two representations: without such a compromise the function will not be evaluable within practical resource limits. If the function is harder than "semihard," i.e. is actually "hard," then no amount of compromise can ever make feasible its evaluation by any terrestrial device.