The loss function is explained by the game theory, the logic is clear and the formula is simple and clear. Thus avoid the use of maximum likelihood estimation to explain the loss function and eliminate the need for Monte Carlo methods to solve the normalized denominator. Assuming that the bridge-type population follows a Boltzmann distribution, a neural network is constructed to represent the energy function. Use Langevin dynamics technology to generate a new sample with low energy value, thus a generative model of bridge-type based on energy is established. Train energy function on symmetric structured image dataset of three span beam bridge, arch bridge, cable-stayed bridge, and suspension bridge to accurately calculate the energy values of real and fake samples. Sampling from latent space, using gradient descent algorithm, the energy function transforms the sampling points into low energy score samples, thereby generating new bridge types different from the dataset. Due to unstable and slow training in this attempt, the possibility of generating new bridge types is rare and the image definition of generated images is low.

This study presents a Bayesian maximum \textit{a~posteriori} (MAP) framework for dynamical system identification from time-series data. This is shown to be equivalent to a generalized zeroth-order Tikhonov regularization, providing a rational justification for the choice of the residual and regularization terms, respectively, from the negative logarithms of the likelihood and prior distributions. In addition to the estimation of model coefficients, the Bayesian interpretation gives access to the full apparatus for Bayesian inference, including the ranking of models, the quantification of model uncertainties and the estimation of unknown (nuisance) hyperparameters. Two Bayesian algorithms, joint maximum \textit{a~posteriori} (JMAP) and variational Bayesian approximation (VBA), are compared to the popular SINDy algorithm for thresholded least-squares regression, by application to several dynamical systems with added noise. For multivariate Gaussian likelihood and prior distributions, the Bayesian formulation gives Gaussian posterior and evidence distributions, in which the numerator terms can be expressed in terms of the Mahalanobis distance or ``Gaussian norm'' $||\vy-\hat{\vy}||^2_{M^{-1}} = (\vy-\hat{\vy})^\top {M^{-1}} (\vy-\hat{\vy})$, where $\vy$ is a vector variable, $\hat{\vy}$ is its estimator and $M$ is the covariance matrix. The posterior Gaussian norm is shown to provide a robust metric for quantitative model selection.

Machine learning models traditionally assume that training and test data are independently and identically distributed. However, in real-world applications, the test distribution often differs from training. This problem, known as out-of-distribution generalization, challenges conventional models. Invariant Risk Minimization (IRM) emerges as a solution, aiming to identify features invariant across different environments to enhance out-of-distribution robustness. However, IRM's complexity, particularly its bi-level optimization, has led to the development of various approximate methods. Our study investigates these approximate IRM techniques, employing the Expected Calibration Error (ECE) as a key metric. ECE, which measures the reliability of model prediction, serves as an indicator of whether models effectively capture environment-invariant features. Through a comparative analysis of datasets with distributional shifts, we observe that Information Bottleneck-based IRM, which condenses representational information, achieves a balance in improving ECE while preserving accuracy relatively. This finding is pivotal, as it demonstrates a feasible path to maintaining robustness without compromising accuracy. Nonetheless, our experiments also caution against over-regularization, which can diminish accuracy. This underscores the necessity for a systematic approach in evaluating out-of-distribution generalization metrics, one that beyond mere accuracy to address the nuanced interplay between accuracy and calibration.

These approaches fall into four categories: physicconstrained, Over recent decades, advances in mechanics and electronics serial, parallel, and ensemble strategies. In have led to the development of increasingly sophisticated the physic-constrained category, techniques either integrate systems with complex and multi-physics dynamics, exposing physically meaningful features from first principles into limitations in first principle-based representations [17]. ML models or explicitly include physical constraints, such Modeling these advanced systems purely based on domain as boundary conditions, into the loss function (see, e.g., knowledge may inadequately capture the overall system behavior, the working principle of physics-informed neural networks often necessitating the formulation of complex partial (PINN)) [7,?].

ML-based systems are used today in a wide range of areas, and increasingly also in safety-critical domains. Their range of application is growing exponentially. At the same time, more and more experts are warning of the uncertainties and risks associated with the uncontrolled and overly rapid development of AI systems Bengio et al. [22.03.2023]. In general, there is a growing need to provide methods and procedures for testing functioning and quality characteristics of these systems. Various methods currently exist to test and verify ML-based systems, be it formal verification, simulation approaches or classical testing Albarghouthi, Jackson et al., Vasu Singh et al., or new analysis methods in the context of XAI Hoyer et al., Guidotti et al.. The methods aim for providing evidence on the robustness and trustworthiness of the ML models or ML-based system (ML - Machine Learning). Similar to the traditional development of complex software systems, testing has also proven to be the most effective method for proving quality and gaining trust in ML.

Investors and regulators are concerned about financial market volatility and crashes. For this reason, the Volatility index (VIX) was introduced in 1993 by the Chicago Board Options Exchange (CBOE) with the aim of assessing the expected financial market volatility in the short-run, i.e. for the next 30 days, since it is calculated as an implied volatility from the options on the S&P 500 index on this time-to-maturity [1]. The VIX has been proven to be a good predictor of expected stock index shifts, and therefore as an early warning for investor sentiment and financial market turbulences (see e.g., [1], and more recently, [2]). Due to its importance for asset managers and regulators, it would be useful to foresee the values of the index; however, the VIX is very difficult to forecast [3]. There exist several proposals to predict time series found in the literature classified as conventional and modern methods (see e.g., [4] and the references therein).

Training machine learning and statistical models often involves optimizing a data-driven risk criterion. The risk is usually computed with respect to the empirical data distribution, but this may result in poor and unstable out-of-sample performance due to distributional uncertainty. In the spirit of distributionally robust optimization, we propose a novel robust criterion by combining insights from Bayesian nonparametric (i.e., Dirichlet Process) theory and recent decision-theoretic models of smooth ambiguity-averse preferences. First, we highlight novel connections with standard regularized empirical risk minimization techniques, among which Ridge and LASSO regressions. Then, we theoretically demonstrate the existence of favorable finite-sample and asymptotic statistical guarantees on the performance of the robust optimization procedure. For practical implementation, we propose and study tractable approximations of the criterion based on well-known Dirichlet Process representations. We also show that the smoothness of the criterion naturally leads to standard gradient-based numerical optimization. Finally, we provide insights into the workings of our method by applying it to high-dimensional sparse linear regression and robust location parameter estimation tasks.

The performance of Markov chain Monte Carlo samplers strongly depends on the properties of the target distribution such as its covariance structure, the location of its probability mass and its tail behavior. We explore the use of bijective affine transformations of the sample space to improve the properties of the target distribution and thereby the performance of samplers running in the transformed space. In particular, we propose a flexible and user-friendly scheme for adaptively learning the affine transformation during sampling. Moreover, the combination of our scheme with Gibbsian polar slice sampling is shown to produce samples of high quality at comparatively low computational cost in several settings based on real-world data.

This report examines a novel risk associated with current (and projected) AI tools. Making effective decisions about future actions requires us to reason under uncertainty (RUU), and doing so is essential to many critical real world problems. Overfaced by this challenge, there is growing demand for AI tools like LLMs to assist decision-makers. Having evidenced this demand and the incentives behind it, we expose a growing risk: we 1) do not currently sufficiently understand LLM capabilities in this regard, and 2) have no guarantees of performance given fundamental computational explosiveness and deep uncertainty constraints on accuracy. This report provides an exposition of what makes RUU so challenging for both humans and machines, and relates these difficulties to prospective AI timelines and capabilities. Having established this current potential misuse risk, we go on to expose how this seemingly additive risk (more misuse additively contributed to potential harm) in fact has multiplicative properties. Specifically, we detail how this misuse risk connects to a wider network of underlying structural risks (e.g., shifting incentives, limited transparency, and feedback loops) to produce non-linear harms. We go on to provide a solutions roadmap that targets multiple leverage points in the structure of the problem. This includes recommendations for all involved actors (prospective users, developers, and policy-makers) and enfolds insights from areas including Decision-making Under Deep Uncertainty and complex systems theory. We argue this report serves not only to raise awareness (and subsequently mitigate/correct) of a current, novel AI risk, but also awareness of the underlying class of structural risks by illustrating how their interconnected nature poses twin-dangers of camouflaging their presence, whilst amplifying their potential effects.

Simulating sampling algorithms with people has proven a useful method for efficiently probing and understanding their mental representations. We propose that the same methods can be used to study the representations of Large Language Models (LLMs). While one can always directly prompt either humans or LLMs to disclose their mental representations introspectively, we show that increased efficiency can be achieved by using LLMs as elements of a sampling algorithm. We explore the extent to which we recover human-like representations when LLMs are interrogated with Direct Sampling and Markov chain Monte Carlo (MCMC). We found a significant increase in efficiency and performance using adaptive sampling algorithms based on MCMC. We also highlight the potential of our method to yield a more general method of conducting Bayesian inference \textit{with} LLMs.