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Data assimilation is crucial in a wide range of applications, but it often faces challenges such as high computational costs due to data dimensionality and incomplete understanding of underlying mechanisms. To address these challenges, this study presents a novel assimilation framework, termed Latent Assimilation with Implicit Neural Representations (LAINR). By introducing Spherical Implicit Neural Representations (SINR) along with a data-driven uncertainty estimator of the trained neural networks, LAINR enhances efficiency in assimilation process. Experimental results indicate that LAINR holds certain advantage over existing methods based on AutoEncoders, both in terms of accuracy and efficiency.

Large language models (LLMs) are becoming increasingly relevant as a potential tool for healthcare, aiding communication between clinicians, researchers, and patients. However, traditional evaluations of LLMs on medical exam questions do not reflect the complexity of real patient-doctor interactions. An example of this complexity is the introduction of patient self-diagnosis, where a patient attempts to diagnose their own medical conditions from various sources. While the patient sometimes arrives at an accurate conclusion, they more often are led toward misdiagnosis due to the patient's over-emphasis on bias validating information. In this work we present a variety of LLMs with multiple-choice questions from United States medical board exams which are modified to include self-diagnostic reports from patients. Our findings highlight that when a patient proposes incorrect bias-validating information, the diagnostic accuracy of LLMs drop dramatically, revealing a high susceptibility to errors in self-diagnosis.

In these pedagogic notes I review the statistical mechanics approach to neural networks, focusing on the paradigmatic example of the perceptron architecture with binary an continuous weights, in the classification setting. I will review the Gardner's approach based on replica method and the derivation of the SAT/UNSAT transition in the storage setting. Then, I discuss some recent works that unveiled how the zero training error configurations are geometrically arranged, and how this arrangement changes as the size of the training set increases. I also illustrate how different regions of solution space can be explored analytically and how the landscape in the vicinity of a solution can be characterized. I give evidence how, in binary weight models, algorithmic hardness is a consequence of the disappearance of a clustered region of solutions that extends to very large distances. Finally, I demonstrate how the study of linear mode connectivity between solutions can give insights into the average shape of the solution manifold.

Complex networks are used to model many real-world systems. However, the dimensionality of these systems can make them challenging to analyze. Dimensionality reduction techniques like POD can be used in such cases. However, these models are susceptible to perturbations in the input data. We propose an algorithmic framework that combines techniques from pattern recognition (PR) and stochastic filtering theory to enhance the output of such models. The results of our study show that our method can improve the accuracy of the surrogate model under perturbed inputs. Deep Neural Networks (DNNs) are susceptible to adversarial attacks. However, recent research has revealed that Neural Ordinary Differential Equations (neural ODEs) exhibit robustness in specific applications. We benchmark our algorithmic framework with the neural ODE-based approach as a reference.

Mediation analysis is widely used in health science research to evaluate the extent to which an intermediate variable explains an observed exposure-outcome relationship. However, the validity of analysis can be compromised when the exposure is measured with error. Motivated by the Health Professionals Follow-up Study (HPFS), we investigate the impact of exposure measurement error on assessing mediation with a survival outcome, based on the Cox proportional hazards outcome model. When the outcome is rare and there is no exposure-mediator interaction, we show that the uncorrected estimators of the natural indirect and direct effects can be biased into either direction, but the uncorrected estimator of the mediation proportion is approximately unbiased as long as the measurement error is not large or the mediator-exposure association is not strong. We develop ordinary regression calibration and risk set regression calibration approaches to correct the exposure measurement error-induced bias when estimating mediation effects and allowing for an exposure-mediator interaction in the Cox outcome model. The proposed approaches require a validation study to characterize the measurement error process. We apply the proposed approaches to the HPFS (1986-2016) to evaluate extent to which reduced body mass index mediates the protective effect of vigorous physical activity on the risk of cardiovascular diseases, and compare the finite-sample properties of the proposed estimators via simulations.

Conventional neural network elastoplasticity models are often perceived as lacking interpretability. This paper introduces a two-step machine-learning approach that returns mathematical models interpretable by human experts. In particular, we introduce a surrogate model where yield surfaces are expressed in terms of a set of single-variable feature mappings obtained from supervised learning. A postprocessing step is then used to re-interpret the set of single-variable neural network mapping functions into mathematical form through symbolic regression. This divide-and-conquer approach provides several important advantages. First, it enables us to overcome the scaling issue of symbolic regression algorithms. From a practical perspective, it enhances the portability of learned models for partial differential equation solvers written in different programming languages. Finally, it enables us to have a concrete understanding of the attributes of the materials, such as convexity and symmetries of models, through automated derivations and reasoning. Numerical examples have been provided, along with an open-source code to enable third-party validation.

This work uses the entropy-regularised relaxed stochastic control perspective as a principled framework for designing reinforcement learning (RL) algorithms. Herein agent interacts with the environment by generating noisy controls distributed according to the optimal relaxed policy. The noisy policies on the one hand, explore the space and hence facilitate learning but, on the other hand, introduce bias by assigning a positive probability to non-optimal actions. This exploration-exploitation trade-off is determined by the strength of entropy regularisation. We study algorithms resulting from two entropy regularisation formulations: the exploratory control approach, where entropy is added to the cost objective, and the proximal policy update approach, where entropy penalises policy divergence between consecutive episodes. We focus on the finite horizon continuous-time linear-quadratic (LQ) RL problem, where a linear dynamics with unknown drift coefficients is controlled subject to quadratic costs. In this setting, both algorithms yield a Gaussian relaxed policy. We quantify the precise difference between the value functions of a Gaussian policy and its noisy evaluation and show that the execution noise must be independent across time. By tuning the frequency of sampling from relaxed policies and the parameter governing the strength of entropy regularisation, we prove that the regret, for both learning algorithms, is of the order $\mathcal{O}(\sqrt{N}) $ (up to a logarithmic factor) over $N$ episodes, matching the best known result from the literature.

Current physics-informed (standard or operator) neural networks still rely on accurately learning the initial conditions of the system they are solving. In contrast, standard numerical methods evolve such initial conditions without needing to learn these. In this study, we propose to improve current physics-informed deep learning strategies such that initial conditions do not need to be learned and are represented exactly in the predicted solution. Moreover, this method guarantees that when a DeepONet is applied multiple times to time step a solution, the resulting function is continuous.

In machine learning models, the estimation of errors is often complex due to distribution bias, particularly in spatial data such as those found in environmental studies. We introduce an approach based on the ideas of importance sampling to obtain an unbiased estimate of the target error. By taking into account difference between desirable error and available data, our method reweights errors at each sample point and neutralizes the shift. Importance sampling technique and kernel density estimation were used for reweighteing. We validate the effectiveness of our approach using artificial data that resemble real-world spatial datasets. Our findings demonstrate advantages of the proposed approach for the estimation of the target error, offering a solution to a distribution shift problem. Overall error of predictions dropped from 7% to just 2% and it gets smaller for larger samples.

Accurately labeling biomedical data presents a challenge. Traditional semi-supervised learning methods often under-utilize available unlabeled data. To address this, we propose a novel reliability-based training data cleaning method employing inductive conformal prediction (ICP). This method capitalizes on a small set of accurately labeled training data and leverages ICP-calculated reliability metrics to rectify mislabeled data and outliers within vast quantities of noisy training data. The efficacy of the method is validated across three classification tasks within distinct modalities: filtering drug-induced-liver-injury (DILI) literature with title and abstract, predicting ICU admission of COVID-19 patients through CT radiomics and electronic health records, and subtyping breast cancer using RNA-sequencing data. Varying levels of noise to the training labels were introduced through label permutation. Results show significant enhancements in classification performance: accuracy enhancement in 86 out of 96 DILI experiments (up to 11.4%), AUROC and AUPRC enhancements in all 48 COVID-19 experiments (up to 23.8% and 69.8%), and accuracy and macro-average F1 score improvements in 47 out of 48 RNA-sequencing experiments (up to 74.6% and 89.0%). Our method offers the potential to substantially boost classification performance in multi-modal biomedical machine learning tasks. Importantly, it accomplishes this without necessitating an excessive volume of meticulously curated training data.