Learning Space-Time Continuous Neural PDEs from Partially Observed States

We introduce a novel grid-independent model for learning partial differential equations (PDEs) from noisy and partial observations on irregular spatiotemporal grids. We propose a space-time continuous latent neural PDE model with an efficient probabilistic framework and a novel encoder design for improved data efficiency and grid independence. The latent state dynamics are governed by a PDE model that combines the collocation method and the method of lines. We employ amortized variational inference for approximate posterior estimation and utilize a multiple shooting technique for enhanced training speed and stability. Our model demonstrates state-of-the-art performance on complex synthetic and real-world datasets, overcoming limitations of previous approaches and effectively handling partially-observed data. The proposed model outperforms recent methods, showing its potential to advance data-driven PDE modeling and enabling robust, grid-independent modeling of complex partially-observed dynamic processes

Modeling binding specificities of transcription factor pairs with random forests

Background Transcription factors (TFs) bind regulatory DNA regions with sequence specificity, form complexes and regulate gene expression. In cooperative TF-TF binding, two transcription factors bind onto a shared DNA binding site as a pair. Previous work has demonstrated pairwise TF-TF-DNA interactions with position weight matrices (PWMs), which may however not sufficiently take into account the complexity and flexibility of pairwise binding. Results We propose two random forest (RF) methods for joint TF-TF binding site prediction: ComBind and JointRF. We train models with previously published large-scale CAP-SELEX DNA libraries, which comprise DNA sequences enriched for binding of a selected TF pair. JointRF builds a random forest with sub-sequences selected from CAP-SELEX DNA reads with previously proposed pairwise PWM. JointRF outperforms (area under receiver operating characteristics curve, AUROC, 0.75) the current state-of-the-art method i.e. orientation and spacing specific pairwise PWMs (AUROC 0.59). Thus, JointRF may be utilized to improve prediction accuracy for pre-determined binding preferences. However, pairwise TF binding is currently considered flexible; a pair may bind DNA with different orientations and amounts of dinucleotide gaps or overlap between the two motifs. Thus, we developed ComBind, which utilizes random forests by considering simultaneously multiple orientations and spacings of the two factors. Our approach outperforms (AUROC 0.78) PWMs, as well as JointRF (pPeer reviewe

A Variational Autoencoder for Heterogeneous Temporal and Longitudinal Data

The variational autoencoder (VAE) is a popular deep latent variable model used to analyse high-dimensional datasets by learning a low-dimensional latent representation of the data. It simultaneously learns a generative model and an inference network to perform approximate posterior inference. Recently proposed extensions to VAEs that can handle temporal and longitudinal data have applications in healthcare, behavioural modelling, and predictive maintenance. However, these extensions do not account for heterogeneous data (i.e., data comprising of continuous and discrete attributes), which is common in many real-life applications. In this work, we propose the heterogeneous longitudinal VAE (HL-VAE) that extends the existing temporal and longitudinal VAEs to heterogeneous data. HL-VAE provides efficient inference for high-dimensional datasets and includes likelihood models for continuous, count, categorical, and ordinal data while accounting for missing observations. We demonstrate our model's efficacy through simulated as well as clinical datasets, and show that our proposed model achieves competitive performance in missing value imputation and predictive accuracy.Comment: Preprin

Latent Neural ODEs with Sparse Bayesian Multiple Shooting

Training dynamic models, such as neural ODEs, on long trajectories is a hard problem that requires using various tricks, such as trajectory splitting, to make model training work in practice. These methods are often heuristics with poor theoretical justifications, and require iterative manual tuning. We propose a principled multiple shooting technique for neural ODEs that splits the trajectories into manageable short segments, which are optimised in parallel, while ensuring probabilistic control on continuity over consecutive segments. We derive variational inference for our shooting-based latent neural ODE models and propose amortized encodings of irregularly sampled trajectories with a transformer-based recognition network with temporal attention and relative positional encoding. We demonstrate efficient and stable training, and state-of-the-art performance on multiple large-scale benchmark datasets

Bayesian inference of ODEs with Gaussian processes

Recent machine learning advances have proposed black-box estimation of unknown continuous-time system dynamics directly from data. However, earlier works are based on approximative ODE solutions or point estimates. We propose a novel Bayesian nonparametric model that uses Gaussian processes to infer posteriors of unknown ODE systems directly from data. We derive sparse variational inference with decoupled functional sampling to represent vector field posteriors. We also introduce a probabilistic shooting augmentation to enable efficient inference from arbitrarily long trajectories. The method demonstrates the benefit of computing vector field posteriors, with predictive uncertainty scores outperforming alternative methods on multiple ODE learning tasks