Implementing neural network mixed-effects models in Template Model Builder (TMB)

Neural network mixed-effects models (NMMs) have gained traction by combining the strong representation and predictive power of artificial neural networks with the capacity of mixed-effects modeling to capture complex correlation structures. However, existing estimation approaches rely heavily on manual derivations of objective functions and gradients, which inherently forces simplifying approximations and severely constrains the complexity and accuracy of NMMs. In this work, we introduce a general framework for implementing NMMs using Template Model Builder (TMB). By leveraging automatic differentiation and Laplace approximation, TMB requires users to specify only the negative joint log-likelihood and any regularization terms. The framework automatically integrates out random effects and evaluates the marginal objective function alongside its exact gradients, eliminating the need for manual derivations or ad hoc approximations. We demonstrate the efficiency, flexibility, and statistical performance of TMB-based NMMs across two numerical examples, including an application to monotonic NMMs. Reproducible code is provided to facilitate broader adoption.

Adaptive KappaSharp: Condition-Number Shaping for Preferential Bayesian Optimization

Preferential Bayesian optimization (PBO) optimizes objectives accessible only through pairwise user comparisons. The standard approach fits a Gaussian process surrogate for observed pairwise comparisons (PairwiseGP) using the Laplace approximation and selects queries with the Expected Utility of Best Option (EUBO) acquisition function. EUBO queries new candidates at each step, producing pairs that share no candidates with previous queries. Each such pair forms an isolated component in the comparison graph, removing one degree of freedom from the likelihood Hessian and making it rank-deficient. This deficiency is structural and cannot be resolved by changing the surrogate modeling approach. Existing approaches to remedy this issue either waste query budget by forcing comparisons to stay connected, or apply uniform regularization that also perturbs directions already well-constrained by the observed comparisons. We propose KappaSharp that enables a diagonal correction to the Hessian to reduce its condition number, with larger corrections where the prior uncertainty is higher. The correction is only applied in the model fitting step, not query selection. An adaptive variant of KappaSharp is also presented that activates the correction only when the surrogate is confident about recent comparisons, avoiding unnecessary corrections when the problem is well-conditioned. On 11 benchmarks (5--20 dimensions), including a 16-dimensional controller tuning problem in plasma medicine, Adaptive KappaSharp outperforms the standard PBO baseline, with up to +10.9% ($p{=}0.003$).

Sequential sparse Gaussian process quantile regression

Quantile regression aims to estimate the conditional quantiles of a response variable from observed data. In a Bayesian setting, Gaussian process quantile regression provides uncertainty quantification but faces significant computational challenges due to the nonconjugacy of the asymmetric Laplace likelihood and the cost of posterior inference. We develop a sparse Gaussian process framework in which the quantile function is represented through a reduced set of inducing variables and posterior inference is performed using a Laplace approximation. A decomposition of the predictive uncertainty into conditional-prior and posterior-induced variance components is then exploited to drive two complementary adaptive mechanisms: inducing-input infilling and data acquisition. These mechanisms are combined within a sequential algorithm that allocates computational effort toward the dominant source of predictive uncertainty and adaptively controls model complexity. Numerical experiments on benchmark problems demonstrate the accuracy of the Laplace approximation, the benefits of variance-based inducing-input placement, and the effectiveness of the proposed sequential enrichment strategy compared with predefined data-acquisition strategies.

Amortized Probabilistic Retrieval of Atmospheric CO2 from OCO-2 Spectra Using Deep Learning with Laplace Approximations and Normalizing Flows

Space-based monitoring of atmospheric carbon dioxide (CO2) is essential for constraining the global carbon budget. NASA's Orbiting Carbon Observatory-2 (OCO-2) estimates column-averaged dry-air mole fractions of CO2 (XCO2) using high-resolution spectra. However, current operational retrieval algorithms are computationally expensive and do not properly quantify uncertainties. We present a novel deep learning framework that addresses these challenges. Due to the difficulties of ground-truth data for real satellite observations, we develop and validate our approach using a high-fidelity simulation dataset. This dataset, created to support OCO-2 uncertainty quantification (UQ), incorporates realistic forward model errors. Our architecture encodes spectral bands using a multi-branch neural network and estimates posteriors of the full CO2 column or desired summaries thereof using two scalable UQ methods: Laplace approximations and normalizing flows. Our approach has five key advantages relative to operational "full-physics" solvers: (1) Amortization: Inference is orders of magnitude faster, enabling real-time processing of massive data streams; (2) Model error robustness: By training on simulations that explicitly include model discrepancies, our method accounts for systematic errors often neglected by standard inversions; (3) Point estimate accuracy: We achieve superior predictive accuracy compared to baseline methods; (4) Improved UQ: The probabilistic outputs yield better-calibrated uncertainty estimates; and (5) Non-Gaussian posteriors: When utilizing normalizing flows, our framework successfully models complex, asymmetric posterior distributions, overcoming the limitations of the Gaussian assumption. These results suggest that simulation-based deep learning is a viable path toward next-generation operational processing systems.

Uncertainty Estimation for Molecular Diffusion Models

Diffusion models have seen wide adoption for 3D molecular generation, yet they offer no principled signal of when a generated molecule is likely to be of low quality. We propose a post-hoc method for estimating per-sample uncertainty in pretrained molecular diffusion models. Building on a Laplace approximation of the denoising network, we measure the variability of the noise prediction across the generation trajectory. Empirically, we show that the resulting uncertainty score is informative of sample quality, exhibiting a negative correlation with established sample-level quality metrics. We further study how the proposed uncertainty score can be used to filter generated samples, improving model performance via test-time scaling.

Instance-Level Post Hoc Uncertainty Quantification in Object Detection

Object detection is a safety-critical component of autonomous driving. It is essential to quantify the uncertainty in bounding-box predictions for safety assurance. Post hoc uncertainty quantification without retraining aligns with real-world deployment requirements; therefore, we employ the Laplace approximation. Because instance-level uncertainty is needed, linearized inference methods that require multiple backpropagations are not time-efficient, and sampling-based methods are not fully post hoc. We propose Monte-Carlo generalized linearized model (MC-GLM), which provides instance-level and approximately post hoc uncertainty quantification. The number of samples required in the Monte Carlo step is constant and independent of the number of output instances, so it can be parallelized. Experiments on the nuScenes dataset with the CenterPoint detector validate the effectiveness of our method, and the resulting uncertainties exhibit good quality.

Laplace Approximation for Bayesian Tensor Network Kernel Machines

Uncertainty estimation is essential for robust decision-making in the presence of ambiguous or out-of-distribution inputs. Gaussian Processes (GPs) are classical kernel-based models that offer principled uncertainty quantification and perform well on small- to medium-scale datasets. Alternatively, formulating the weight space learning problem under tensor network assumptions yields scalable tensor network kernel machines. However, these assumptions break Gaussianity, complicating standard probabilistic inference. This raises a fundamental question: how can tensor network kernel machines provide principled uncertainty estimates? We propose a novel Bayesian Tensor Network Kernel Machine (LA-TNKM) that employs a (linearized) Laplace approximation for Bayesian inference. A comprehensive set of numerical experiments shows that the proposed method consistently matches or surpasses Gaussian Processes and Bayesian Neural Networks (BNNs) across diverse UCI regression benchmarks, highlighting both its effectiveness and practical relevance.