We present generative artificial intelligence (AI) to empirically validate fundamental laws of physics, focusing on the Stefan-Boltzmann law linking stellar temperature and luminosity. Our approach simulates counterfactual luminosities under hypothetical temperature regimes for each individual star and iteratively refines the temperature-luminosity relationship in a deep learning architecture. We use Gaia DR3 data and find that, on average, temperature's effect on luminosity increases with stellar radius and decreases with absolute magnitude, consistent with theoretical predictions. By framing physics laws as causal problems, our method offers a novel, data-driven approach to refine theoretical understanding and inform evidence-based policy and practice.

Kolmogorov GAM (K-GAM) networks are shown to be an efficient architecture for training and inference. They are an additive model with an embedding that is independent of the function of interest. They provide an alternative to the transformer architecture. They are the machine learning version of Kolmogorov's Superposition Theorem (KST) which provides an efficient representations of a multivariate function. Such representations have use in machine learning for encoding dictionaries (a.k.a. "look-up" tables). KST theory also provides a representation based on translates of the K\"oppen function. The goal of our paper is to interpret this representation in a machine learning context for applications in Artificial Intelligence (AI). Our architecture is equivalent to a topological embedding which is independent of the function together with an additive layer that uses a Generalized Additive Model (GAM). This provides a class of learning procedures with far fewer parameters than current deep learning algorithms. Implementation can be parallelizable which makes our algorithms computationally attractive. To illustrate our methodology, we use the Iris data from statistical learning. We also show that our additive model with non-linear embedding provides an alternative to transformer architectures which from a statistical viewpoint are kernel smoothers. Additive KAN models therefore provide a natural alternative to transformers. Finally, we conclude with directions for future research.

Generative methods (Gen-AI) are reviewed with a particular goal to solving tasks in Machine Learning and Bayesian inference. Generative models require one to simulate a large training dataset and to use deep neural networks to solve a supervised learning problem. To do this, we require high dimensional regression methods and tools for dimensionality reduction (a.k.a feature selection). The main advantage of Gen-AI methods is their ability to be model-free and to use deep neural networks to estimate conditional densities or posterior quantiles of interest. To illustrate generative methods, we analyze the well-known Ebola data-set. Finally, we conclude with directions for future research.

Our goal is to provide a review of deep learning methods which provide insight into structured high-dimensional data. Rather than using shallow additive architectures common to most statistical models, deep learning uses layers of semi-affine input transformations to provide a predictive rule. Applying these layers of transformations leads to a set of attributes (or, features) to which probabilistic statistical methods can be applied. Thus, the best of both worlds can be achieved: scalable prediction rules fortified with uncertainty quantification, where sparse regularization finds the features. Deep learning is one of the widely used machine learning method for analysis of large scale and highdimensional data sets.

We propose a neural network-based approach to calculate the value of a chess square-piece combination. Our model takes a triplet (Color, Piece, Square) as an input and calculates a value that measures the advantage/disadvantage of having this piece on this square. Our methods build on recent advances in chess AI, and can accurately assess the worth of positions in a game of chess. The conventional approach assigns fixed values to pieces $(\symking=\infty, \symqueen=9, \symrook=5, \symbishop=3, \symknight=3, \sympawn=1)$. We enhance this analysis by introducing marginal valuations. We use deep Q-learning to estimate the parameters of our model. We demonstrate our method by examining the positioning of Knights and Bishops, and also provide valuable insights into the valuation of pawns. Finally, we conclude by suggesting potential avenues for future research.

Quantum Bayesian Computation (QBC) is an emerging field that levers the computational gains available from quantum computers to provide an exponential speed-up in Bayesian computation. Our paper adds to the literature in two ways. First, we show how von Neumann quantum measurement can be used to simulate machine learning algorithms such as Markov chain Monte Carlo (MCMC) and Deep Learning (DL) that are fundamental to Bayesian learning. Second, we describe data encoding methods needed to implement quantum machine learning including the counterparts to traditional feature extraction and kernel embeddings methods. Our goal then is to show how to apply quantum algorithms directly to statistical machine learning problems. On the theoretical side, we provide quantum versions of high dimensional regression, Gaussian processes (Q-GP) and stochastic gradient descent (Q-SGD). On the empirical side, we apply a Quantum FFT model to Chicago housing data. Finally, we conclude with directions for future research.

Transportation activity-based simulators (ABMs) represent an individual traveler's activity patterns and trips throughout the day by using nested choice models. The generated trips are then simulated in a traffic flow simulator to learn system-level patterns. These behaviorally-realistic models require a high-resolution representation of network flows and, thus, are computationally expensive. The very same flexibility which makes these simulation models appealing, also makes their calibration problems intractable, with the number of simulations required to find an optimal solution growing exponentially as the input dimension increases [90, 70]. As a result, the use of these simulators is currently limited to what-if analysis. This paper focuses on calibrating the static choice model parameters used in activity-based simulators. The goal of calibration is to find values of the simulator's input parameters θ that minimizes the deviance between observed data and simulator's outputs.

Merging the two cultures of deep and statistical learning provides insights into structured high-dimensional data. Traditional statistical modeling is still a dominant strategy for structured tabular data. Deep learning can be viewed through the lens of generalized linear models (GLMs) with composite link functions. Sufficient dimensionality reduction (SDR) and sparsity performs nonlinear feature engineering. We show that prediction, interpolation and uncertainty quantification can be achieved using probabilistic methods at the output layer of the model. Thus a general framework for machine learning arises that first generates nonlinear features (a.k.a factors) via sparse regularization and stochastic gradient optimisation and second uses a stochastic output layer for predictive uncertainty. Rather than using shallow additive architectures as in many statistical models, deep learning uses layers of semi affine input transformations to provide a predictive rule. Applying these layers of transformations leads to a set of attributes (a.k.a features) to which predictive statistical methods can be applied. Thus we achieve the best of both worlds: scalability and fast predictive rule construction together with uncertainty quantification. Sparse regularisation with un-supervised or supervised learning finds the features. We clarify the duality between shallow and wide models such as PCA, PPR, RRR and deep but skinny architectures such as autoencoders, MLPs, CNN, and LSTM. The connection with data transformations is of practical importance for finding good network architectures. By incorporating probabilistic components at the output level we allow for predictive uncertainty. For interpolation we use deep Gaussian process and ReLU trees for classification. We provide applications to regression, classification and interpolation. Finally, we conclude with directions for future research.

In this paper, we present a deep learning technique-based method to solve large-scale 0-1 knapsack problems where the number of products (items) is large and/or the values of products are not necessarily predetermined but decided by an external value assignment function during the optimization process. Our solution is greatly inspired by the method of Lagrange multiplier and some recent adoptions of game theory to deep learning. After formally defining our proposed method based on them, we develop an adaptive gradient ascent method to stabilize its optimization process. In our experiments, the presented method solves all the large-scale benchmark KP instances in about a minute, whereas existing methods show fluctuating runtime. We also show that our method can be used for other applications, including but not limited to the point cloud resampling.

In this article we review computational aspects of Deep Learning (DL). Deep learning uses network architectures consisting of hierarchical layers of latent variables to construct predictors for high-dimensional input-output models. Training a deep learning architecture is computationally intensive, and efficient linear algebra libraries is the key for training and inference. Stochastic gradient descent (SGD) optimization and batch sampling are used to learn from massive data sets.