We propose the NeRF-LEBM, a likelihood-based top-down 3D-aware 2D image generative model that incorporates 3D representation via Neural Radiance Fields (NeRF) and 2D imaging process via differentiable volume rendering. The model represents an image as a rendering process from 3D object to 2D image and is conditioned on some latent variables that account for object characteristics and are assumed to follow informative trainable energy-based prior models. We propose two likelihood-based learning frameworks to train the NeRF-LEBM: (i) maximum likelihood estimation with Markov chain Monte Carlo-based inference and (ii) variational inference with the reparameterization trick. We study our models in the scenarios with both known and unknown camera poses. Experiments on several benchmark datasets demonstrate that the NeRF-LEBM can infer 3D object structures from 2D images, generate 2D images with novel views and objects, learn from incomplete 2D images, and learn from 2D images with known or unknown camera poses.

Solar energy has been developed as a better alternative to fossil fuels in the past few years. It is a renewable and infinite source of energy which does not have a bad impact on the environment. It is also cheap and easily accessible, making it a better alternative for both personal and commercial purposes. Solar Arrays are made when PV modules used in solar panels are connected together. Energy is produced when sunlight falls on Solar Panels which can be used instead of Fossil fuel's produced energy. For execution of a PV system under different situations, estimating the parameters in a PV model plays an important role because it enables us to optimise the design and performance of the system which leads to increased energy production and improved performance. If a PV system is not performing as expected, then identification of parameters of the PV model helps identify the root cause of the problem. This could be due to factors such as shading, module mismatch, or degradation over time. By accurately estimating the parameters, we can determine the best method to improve its performance.

In today's rapidly advancing world of Artificial Intelligence (AI), the need for explainable AI has become more critical than ever. As AI systems are being increasingly integrated into various aspects of our daily lives, it is crucial to understand how these systems make decisions and provide explanations for their actions. Bayesian networks, a powerful and versatile graphical modeling technique, are gaining prominence as a tool for building explainable AI models. In this blog, we will demystify Bayesian networks and explore their relevance in the field of AI. We will delve into the fundamentals of Bayesian networks, their applications in AI, and how they enable explainable AI.

Minimum miscibility pressure (MMP) prediction plays an important role in design and operation of nitrogen based enhanced oil recovery processes. In this work, a comparative study of statistical and machine learning methods used for MMP estimation is carried out. Most of the predictive models developed in this study exhibited superior performance over correlation and predictive models reported in literature.

Inverse problems are ubiquitous in the sciences and engineering. Two categories of inverse problems concerning a physical system are (1) estimate parameters in a model of the system from observed input-output pairs and (2) given a model of the system, reconstruct the input to it that caused some observed output. Applied inverse problems are challenging because a solution may (i) not exist, (ii) not be unique, or (iii) be sensitive to measurement noise contaminating the data. Bayesian statistical inversion (BSI) is an approach to tackle ill-posed and/or ill-conditioned inverse problems. Advantageously, BSI provides a "solution" that (i) quantifies uncertainty by assigning a probability to each possible value of the unknown parameter/input and (ii) incorporates prior information and beliefs about the parameter/input. Herein, we provide a tutorial of BSI for inverse problems, by way of illustrative examples dealing with heat transfer from ambient air to a cold lime fruit. First, we use BSI to infer a parameter in a dynamic model of the lime temperature from measurements of the lime temperature over time. Second, we use BSI to reconstruct the initial condition of the lime from a measurement of its temperature later in time. We demonstrate the incorporation of prior information, visualize the posterior distributions of the parameter/initial condition, and show posterior samples of lime temperature trajectories from the model. Our tutorial aims to reach a wide range of scientists and engineers.

The framework of causal models provides a principled approach to causal reasoning, applied today across many scientific domains. Here we present this framework in the language of string diagrams, interpreted formally using category theory. A class of string diagrams, called network diagrams, are in 1-to-1 correspondence with directed acyclic graphs. A causal model is given by such a diagram with its components interpreted as stochastic maps, functions, or general channels in a symmetric monoidal category with a 'copy-discard' structure (cd-category), turning a model into a single mathematical object that can be reasoned with intuitively and yet rigorously. Building on prior works by Fong and Jacobs, Kissinger and Zanasi, as well as Fritz and Klingler, we present diagrammatic definitions of causal models and functional causal models in a cd-category, generalising causal Bayesian networks and structural causal models, respectively. We formalise general interventions on a model, including but beyond do-interventions, and present the natural notion of an open causal model with inputs. We also give an approach to conditioning based on a normalisation box, allowing for causal inference calculations to be done fully diagrammatically. We define counterfactuals in this setup, and treat the problems of the identifiability of causal effects and counterfactuals fully diagrammatically. The benefits of such a presentation of causal models lie in foundational questions in causal reasoning and in their clarificatory role and pedagogical value. This work aims to be accessible to different communities, from causal model practitioners to researchers in applied category theory, and discusses many examples from the literature for illustration. Overall, we argue and demonstrate that causal reasoning according to the causal model framework is most naturally and intuitively done as diagrammatic reasoning.

We study mean-field variational Bayesian inference using the TAP approach, for Z2-synchronization as a prototypical example of a high-dimensional Bayesian model. We show that for any signal strength $\lambda > 1$ (the weak-recovery threshold), there exists a unique local minimizer of the TAP free energy functional near the mean of the Bayes posterior law. Furthermore, the TAP free energy in a local neighborhood of this minimizer is strongly convex. Consequently, a natural-gradient/mirror-descent algorithm achieves linear convergence to this minimizer from a local initialization, which may be obtained by a constant number of iterates of Approximate Message Passing (AMP). This provides a rigorous foundation for variational inference in high dimensions via minimization of the TAP free energy. We also analyze the finite-sample convergence of AMP, showing that AMP is asymptotically stable at the TAP minimizer for any $\lambda > 1$, and is linearly convergent to this minimizer from a spectral initialization for sufficiently large $\lambda$. Such a guarantee is stronger than results obtainable by state evolution analyses, which only describe a fixed number of AMP iterations in the infinite-sample limit. Our proofs combine the Kac-Rice formula and Sudakov-Fernique Gaussian comparison inequality to analyze the complexity of critical points that satisfy strong convexity and stability conditions within their local neighborhoods.

In order to overcome the limitation of low efficiency of existing granularity reduction algorithms in multi-granulation rough sets, based on matrix method, a fast granularity reduction algorithm is proposed and the time complexity is \(O({ U } {2} \cdot A U \cdot { A } {2})\). First, the definitions of positive region matrix and granularity column matrix of multi-granulation space are proposed. Second, through the quantity product of these two matrices, the definition of positive region column matrix is presented. Based on the positive region column matrix, cut matrix and matrix norm are defined, respectively. Third, the matrix-based calculation methods of multi-granulation approximation quality and granularity significance are proposed.

Physics-informed neural networks (PINNs) have proven a suitable mathematical scaffold for solving inverse ordinary (ODE) and partial differential equations (PDE). Typical inverse PINNs are formulated as soft-constrained multi-objective optimization problems with several hyperparameters. In this work, we demonstrate that inverse PINNs can be framed in terms of maximum-likelihood estimators (MLE) to allow explicit error propagation from interpolation to the physical model space through Taylor expansion, without the need of hyperparameter tuning. We explore its application to high-dimensional coupled ODEs constrained by differential algebraic equations that are common in transient chemical and biological kinetics. Furthermore, we show that singular-value decomposition (SVD) of the ODE coupling matrices (reaction stoichiometry matrix) provides reduced uncorrelated subspaces in which PINNs solutions can be represented and over which residuals can be projected. Finally, SVD bases serve as preconditioners for the inversion of covariance matrices in this hyperparameter-free robust application of MLE to ``kinetics-informed neural networks''.

Diffusion probabilistic models have been successful in generating high-quality and diverse images. However, traditional models, whose input and output are high-resolution images, suffer from excessive memory requirements, making them less practical for edge devices. Previous approaches for generative adversarial networks proposed a patch-based method that uses positional encoding and global content information. Nevertheless, designing a patch-based approach for diffusion probabilistic models is non-trivial. In this paper, we resent a diffusion probabilistic model that generates images on a patch-by-patch basis. We propose two conditioning methods for a patch-based generation. First, we propose position-wise conditioning using one-hot representation to ensure patches are in proper positions. Second, we propose Global Content Conditioning (GCC) to ensure patches have coherent content when concatenated together. We evaluate our model qualitatively and quantitatively on CelebA and LSUN bedroom datasets and demonstrate a moderate trade-off between maximum memory consumption and generated image quality. Specifically, when an entire image is divided into 2 x 2 patches, our proposed approach can reduce the maximum memory consumption by half while maintaining comparable image quality.