Implicit and explicit treatments of model error in numerical simulation

Numerical simulations of physical systems exhibit discrepancies arising from unmodeled physics and idealizations, as well as numerical approximation errors stemming from discretization and solver tolerances. This article reviews techniques developed in the past several decades to approximate and account for model errors, both implicitly and explicitly. Beginning from fundamentals, we frame model error in inverse problems, data assimilation, and predictive modeling contexts. We then survey major approaches: the Bayesian approximation error framework, embedded internal error models for structural uncertainty, probabilistic numerical methods for discretization uncertainty, model discrepancy modeling in Bayesian calibration and its recent extensions, machine-learning-based discrepancy correction, multi-fidelity and hybrid modeling strategies, as well as residual-based, variational, and adjoint-driven error estimators. Throughout, we emphasize the conceptual underpinnings of implicit versus explicit error treatment and highlight how these methods improve predictive performance and uncertainty quantification in practical applications ranging from engineering design to Earth-system science. Each section provides an overview of key developments with an extensive list of references to facilitate further reading. The review is written for practitioners of large-scale computational physics and engineering simulation, emphasizing how these methods can be incorporated into PDE solvers, inverse problem workflows, and data assimilation systems.

💡 Research Summary

This review provides a comprehensive taxonomy of model error treatment in large‑scale numerical simulations, distinguishing between implicit and explicit strategies. Model error is first decomposed into structural (model‑form) error and numerical (discretization/solver) error, and the authors clarify how these components interact and can be confounded in practice. Implicit approaches are those that do not alter the governing equations but instead inflate uncertainty representations so that the effect of the error is absorbed statistically. Three major implicit families are surveyed. The Bayesian Approximation Error (BAE) framework treats the discrepancy between a high‑fidelity and a reduced‑fidelity forward model as a random variable with a mean and covariance estimated from offline paired runs; this corrected error term is then folded into the observation model, yielding more realistic posterior distributions for the parameters. Probabilistic numerics re‑interprets deterministic numerical algorithms (ODE solvers, quadrature, linear solvers) as Bayesian inference problems, returning a posterior distribution that quantifies discretization uncertainty and can be propagated through downstream inference or optimization. Finally, data‑assimilation methods often model model error as an additional noise term, estimating its covariance alongside the state covariance to improve filter and smoother performance.

Explicit approaches introduce an auxiliary discrepancy function or operator that directly represents the error. In Bayesian calibration, a discrepancy function δ˜(x) is added to the observation model and assigned a prior (commonly a Gaussian process); learning this function from data captures both structural and numerical deficiencies but requires careful prior design. Machine‑learning‑based correctors (e.g., neural networks, graph networks) learn a mapping from inputs or states to the error, providing real‑time correction of physics‑based solvers. Multi‑fidelity and hybrid modeling combine cheap low‑fidelity models with high‑fidelity corrections, effectively embedding the error as a learned term that bridges fidelity levels. Residual‑, variational‑, and adjoint‑driven error estimators compute the error directly from PDE residuals or adjoint sensitivities, enabling spatial‑temporal localization of the error and adaptive mesh refinement or optimization feedback.

The paper discusses the strengths and limitations of each method, noting that implicit techniques are computationally convenient but may miss structured, non‑Gaussian error features, while explicit methods offer richer correction capabilities at the cost of added model complexity and data requirements. The authors illustrate applications across fluid and solid mechanics, geophysics, and climate‑weather modeling, and they emphasize that hybrid workflows—e.g., combining BAE with probabilistic numerics and machine‑learning correctors—can exploit the complementary advantages of both paradigms. Current challenges highlighted include scalable estimation of high‑dimensional error statistics, handling non‑Gaussian discrepancies, real‑time adaptive correction, and efficient parallel implementation. The review concludes with a forward‑looking perspective on integrating these techniques into modern PDE solvers, inverse‑problem pipelines, and data‑assimilation systems to achieve more reliable predictions and robust uncertainty quantification.