D point given a shape latent code.

Uncertainty quantification (UQ) in scientific machine learning is increasingly critical as neural networks are widely adopted to tackle complex problems across diverse scientific disciplines. For physics-informed neural networks (PINNs), a prominent model in scientific machine learning, uncertainty is typically quantified using Bayesian or dropout methods. However, both approaches suffer from a fundamental limitation: the prior distribution or dropout rate required to construct honest confidence sets cannot be determined without additional information. In this paper, we propose a novel method within the framework of extended fiducial inference (EFI) to provide rigorous uncertainty quantification for PINNs. The proposed method leverages a narrow-neck hyper-network to learn the parameters of the PINN and quantify their uncertainty based on imputed random errors in the observations. This approach overcomes the limitations of Bayesian and dropout methods, enabling the construction of honest confidence sets based solely on observed data. This advancement represents a significant breakthrough for PINNs, greatly enhancing their reliability, interpretability, and applicability to real-world scientific and engineering challenges. Moreover, it establishes a new theoretical framework for EFI, extending its application to large-scale models, eliminating the need for sparse hyper-networks, and significantly improving the automaticity and robustness of statistical inference.

In this paper we propose and investigate a wide class of Mirror Descent updates (MD) and associated novel Generalized Exponentiated Gradient (GEG) algorithms by exploiting various trace-form entropies and associated deformed logarithms and their inverses - deformed (generalized) exponential functions. The proposed algorithms can be considered as extension of entropic MD and generalization of multiplicative updates. In the literature, there exist nowadays over fifty mathematically well defined generalized entropies, so impossible to exploit all of them in one research paper. So we focus on a few selected most popular entropies and associated logarithms like the Tsallis, Kaniadakis and Sharma-Taneja-Mittal and some of their extension like Tempesta or Kaniadakis-Scarfone entropies. The shape and properties of the deformed logarithms and their inverses are tuned by one or more hyperparameters. By learning these hyperparameters, we can adapt to distribution of training data, which can be designed to the specific geometry of the optimization problem, leading to potentially faster convergence and better performance. The using generalized entropies and associated deformed logarithms in the Bregman divergence, used as a regularization term, provides some new insight into exponentiated gradient descent updates.

Variational Autoencoders (VAEs) and other generative models are widely employed in artificial intelligence to synthesize new data. However, current approaches rely on Euclidean geometric assumptions and statistical approximations that fail to capture the structured and emergent nature of data generation. This paper introduces the Convergent Fusion Paradigm (CFP) theory, a novel geometric framework that redefines data generation by integrating dimensional expansion accompanied by qualitative transformation. By modifying the latent space geometry to interact with emergent high-dimensional structures, CFP theory addresses key challenges such as identifiability issues and unintended artifacts like hallucinations in Large Language Models (LLMs). CFP theory is based on two key conceptual hypotheses that redefine how generative models structure relationships between data and algorithms. Through the lens of CFP theory, we critically examine existing metric-learning approaches. CFP theory advances this perspective by introducing time-reversed metric embeddings and structural convergence mechanisms, leading to a novel geometric approach that better accounts for data generation as a structured epistemic process. Beyond its computational implications, CFP theory provides philosophical insights into the ontological underpinnings of data generation. By offering a systematic framework for high-dimensional learning dynamics, CFP theory contributes to establishing a theoretical foundation for understanding the data-relationship structures in AI. Finally, future research in CFP theory will be led to its implications for fully realizing qualitative transformations, introducing the potential of Hilbert space in generative modeling.

While fiducial inference was widely considered a big blunder by R.A. Fisher, the goal he initially set --`inferring the uncertainty of model parameters on the basis of observations' -- has been continually pursued by many statisticians. To this end, we develop a new statistical inference method called extended Fiducial inference (EFI). The new method achieves the goal of fiducial inference by leveraging advanced statistical computing techniques while remaining scalable for big data. EFI involves jointly imputing random errors realized in observations using stochastic gradient Markov chain Monte Carlo and estimating the inverse function using a sparse deep neural network (DNN). The consistency of the sparse DNN estimator ensures that the uncertainty embedded in observations is properly propagated to model parameters through the estimated inverse function, thereby validating downstream statistical inference. Compared to frequentist and Bayesian methods, EFI offers significant advantages in parameter estimation and hypothesis testing. Specifically, EFI provides higher fidelity in parameter estimation, especially when outliers are present in the observations; and eliminates the need for theoretical reference distributions in hypothesis testing, thereby automating the statistical inference process. EFI also provides an innovative framework for semi-supervised learning.

The success of artificial neural networks (ANNs) hinges greatly on the judicious selection of an activation function, introducing non-linearity into network and enabling them to model sophisticated relationships in data. However, the search of activation functions has largely relied on empirical knowledge in the past, lacking theoretical guidance, which has hindered the identification of more effective activation functions. In this work, we offer a proper solution to such issue. Firstly, we theoretically demonstrate the existence of the worst activation function with boundary conditions (WAFBC) from the perspective of information entropy. Furthermore, inspired by the Taylor expansion form of information entropy functional, we propose the Entropy-based Activation Function Optimization (EAFO) methodology. EAFO methodology presents a novel perspective for designing static activation functions in deep neural networks and the potential of dynamically optimizing activation during iterative training. Utilizing EAFO methodology, we derive a novel activation function from ReLU, known as Correction Regularized ReLU (CRReLU). Experiments conducted with vision transformer and its variants on CIFAR-10, CIFAR-100 and ImageNet-1K datasets demonstrate the superiority of CRReLU over existing corrections of ReLU. Extensive empirical studies on task of large language model (LLM) fine-tuning, CRReLU exhibits superior performance compared to GELU, suggesting its broader potential for practical applications.

We propose $\textit{iterative inversion}$ -- an algorithm for learning an inverse function without input-output pairs, but only with samples from the desired output distribution and access to the forward function. The key challenge is a $\textit{distribution shift}$ between the desired outputs and the outputs of an initial random guess, and we prove that iterative inversion can steer the learning correctly, under rather strict conditions on the function. We apply iterative inversion to learn control. Our input is a set of demonstrations of desired behavior, given as video embeddings of trajectories (without actions), and our method iteratively learns to imitate trajectories generated by the current policy, perturbed by random exploration noise. Our approach does not require rewards, and only employs supervised learning, which can be easily scaled to use state-of-the-art trajectory embedding techniques and policy representations. Indeed, with a VQ-VAE embedding, and a transformer-based policy, we demonstrate non-trivial continuous control on several tasks. Further, we report an improved performance on imitating diverse behaviors compared to reward based methods.

Gait phase-based control is a trending research topic for walking-aid robots, especially robotic lower-limb prostheses. Gait phase estimation is a challenge for gait phase-based control. Previous researches used the integration or the differential of the human's thigh angle to estimate the gait phase, but accumulative measurement errors and noises can affect the estimation results. In this paper, a more robust gait phase estimation method is proposed using a unified form of piecewise monotonic gait phase-thigh angle models for various locomotion modes. The gait phase is estimated from only the thigh angle, which is a stable variable and avoids phase drifting. A Kalman filter-based smoother is designed to further suppress the mutations of the estimated gait phase. Based on the proposed gait phase estimation method, a gait phase-based joint angle tracking controller is designed for a transfemoral prosthesis. The proposed gait estimation method, the gait phase smoother, and the controller are evaluated through offline analysis on walking data in various locomotion modes. And the real-time performance of the gait phase-based controller is validated in an experiment on the transfemoral prosthesis.

In previous posts, we already have studied various ways to create new functions like addition, multiplications, division, and composition. But with this function transformation, we can just obtain rational functions. Let's introduce new techniques to transform functions. A function f is one-one (one to one) if f(a) f(b) whenever a b. An example of one to one function is the identity function f(x) x since every x will have himself as a solution.