[Paper Review] Computer-Aided Derivation of Multi-scale Models: A Rewriting Framework

Bin Yang, Walid Belkhir+1 more|arXiv (Cornell University)|Feb 9, 2013

Advanced Mathematical Modeling in Engineering24 references3 citations

TL;DR

This paper presents a computer-aided framework for deriving multi-scale models using a combination of asymptotic analysis and term rewriting techniques. It enables systematic, reusable generation of homogenized models for elliptic PDEs with periodic or thin structures by encoding derivation steps as rewrite rules, achieving high strategy reusability with minimal modification across extensions like multi-dimensional domains, thin regions, and multi-region configurations.

ABSTRACT

We introduce a framework for computer-aided derivation of multi-scale models. It relies on a combination of an asymptotic method used in the field of partial differential equations with term rewriting techniques coming from computer science. In our approach, a multi-scale model derivation is characterized by the features taken into account in the asymptotic analysis. Its formulation consists in a derivation of a reference model associated to an elementary nominal model, and in a set of transformations to apply to this proof until it takes into account the wanted features. In addition to the reference model proof, the framework includes first order rewriting principles designed for asymptotic model derivations, and second order rewriting principles dedicated to transformations of model derivations. We apply the method to generate a family of homogenized models for second order elliptic equations with periodic coefficients that could be posed in multi-dimensional domains, with possibly multi-domains and/or thin domains.

Motivation & Objective

• To address the lack of systematic, automated methods for deriving multi-scale models in industrial and scientific software.
• To bridge the gap between rigorous asymptotic analysis and practical model derivation by encoding derivation steps as computable rewrite rules.
• To enable the automatic generation of homogenized models for second-order elliptic equations with periodic or thin structures through extensible rewriting strategies.
• To improve reusability and extensibility of model derivation processes by separating reference proofs from transformation strategies.
• To lay the foundation for a general-purpose, computer-aided framework that supports feature-rich model derivations without manual re-derivation for each new case.

Proposed method

• The framework models multi-scale model derivation as a sequence of rewrite rule applications, where each rule corresponds to a step in asymptotic analysis.
• First-order rewrite rules encode mathematical transformations such as scaling, integration by parts, and asymptotic expansion in the context of PDEs.
• Second-order rewrite rules (SO-strategies) govern the composition and application of first-order rules, enabling structured derivation of complex models.
• The reference model derivation is encoded as a skeleton proof, which is then generalized through configurable extensions (e.g., multi-dimensional, thin regions, multi-regions).
• The method supports modular extension by defining new rules and strategies that preserve existing proof structures with minimal modification.
• The framework is validated on a family of homogenized models for elliptic PDEs with periodic coefficients, demonstrating reusability across different geometrical and physical configurations.

Experimental results

Research questions

• RQ1 Can asymptotic model derivation be systematically encoded as a sequence of rewrite rules to enable automation?
• RQ2 How can a reference model derivation be generalized to support multiple physical and geometric features such as periodicity, thin structures, and multi-domains?
• RQ3 To what extent can existing derivation strategies be reused when extending a model to new configurations?
• RQ4 What is the minimal set of new rules and strategies required to extend a model derivation to new classes of problems?
• RQ5 Can the framework support the composition of multiple extensions (e.g., thin + multi-region) without full re-derivation?

Key findings

• The framework successfully encoded the derivation of homogenized models for second-order elliptic PDEs with periodic coefficients across multiple configurations.
• For multi-dimensional extensions, 16.6% of first-order rules and 5% of first-order strategies were modified, indicating high reusability.
• In the thin-region extension, no new first-order rules or strategies were required, demonstrating that the same core strategy applies across different geometries.
• The multi-region extension required only 1 additional first-order rule and minimal strategy changes, confirming the framework’s extensibility.
• Second-order strategies were used sparingly—only 2 in the multi-dimensional case—indicating that high-level composition logic is minimal and reusable.
• The method enables systematic, computer-aided derivation of complex models by generalizing a reference proof through modular, rule-based transformations.