NoLimits.jl: Flexible and Composable Nonlinear Mixed-Effects Modeling in Julia

2026-06-23

AI Summary

This work proposes NoLimits.jl, an open-source Julia framework that overcomes the limited support in existing tools for model structure, inference methods, and random-effects distributions in nonlinear mixed-effects modeling. Leveraging a macro-driven modeling language, NoLimits.jl enables flexible composition of ordinary differential equations, Markov models, and neural networks to construct both observation and latent variable models, while unifying frequentist and Bayesian inference. It is the first open-source framework to offer highly composable modeling with support for covariate-dependent observation and random-effects distributions. The implementation integrates automatic differentiation, normalizing flows, and multiple inference algorithms—including Laplace approximation, stochastic EM, and MCMC. Three case studies demonstrate substantial gains in model expressiveness, flexibility, and inferential capability, significantly expanding the class of nonlinear mixed-effects models that can be specified, estimated, and compared.

Abstract

Nonlinear mixed-effects models are widely used to analyze longitudinal data, but existing open-source software often supports only a limited subset of the model structures, inference methods, machine-learning components, automatic differentiation techniques, and random-effects distributions required in modern applications. We introduce NoLimits.jl, an open-source Julia package for flexible and composable nonlinear mixed-effects modeling. Its macro-based modeling language enables observation and latent-state models to be constructed from diverse building blocks, including ordinary differential equations, Markov models, and neural networks. NoLimits.jl supports flexible, covariate-dependent observation and random-effects distributions and provides a unified interface to frequentist inference through Laplace approximation, stochastic expectation maximization, and Bayesian Markov chain Monte Carlo methods. We demonstrate the package on three case studies showcasing its workflows, integration of differentiable machine-learning components, and data-driven estimation of random-effects distributions using normalizing flows. Together, these capabilities substantially expand the range of nonlinear mixed-effects models that can be specified, estimated, and compared within a single open-source framework.

Problem

• nonlinear mixed-effects models
• flexible modeling
• composable frameworks
• random-effects distributions
• longitudinal data analysis

Innovation

• nonlinear mixed-effects models
• composable modeling
• automatic differentiation
• normalizing flows
• Bayesian inference