Subtopic Deep Dive
Taylor Models for Verified Computing
Research Guide
What is Taylor Models for Verified Computing?
Taylor models represent functions as Taylor series polynomials plus interval remainders for verified enclosures in numerical computing, minimizing wrapping effects in interval arithmetic.
Taylor models combine a multivariate Taylor polynomial with a tight interval remainder bound to enclose function ranges precisely (Neher et al., 2007, 122 citations). They enable validated integration of ODEs and reachability analysis for hybrid systems (Chen et al., 2012, 194 citations). Over 10 key papers since 1996 address applications in optimization and error estimation.
Why It Matters
Taylor models provide tight enclosures for nonlinear ODEs, enabling rigorous verification of hybrid systems safety (Chen et al., 2012). They improve floating-point error bounds in numerical libraries via symbolic Taylor expansions (Solovyev et al., 2018). In global optimization, Taylor models support inner region computations and convex relaxations for parametric systems (Trombettoni et al., 2011; Sahlodin and Chachuat, 2011).
Key Research Challenges
Reducing Wrapping Effects
Interval arithmetic introduces wrapping errors that loosen enclosures during dependency propagation in multivariate Taylor models. Neher et al. (2007) address this in ODE integration by optimizing polynomial degrees. Adaptive remainder bounding remains critical for high dimensions.
Scalable Flowpipe Construction
Building verified flowpipes for nonlinear hybrid systems requires higher-order Taylor models without excessive computation. Chen et al. (2012) propose bounded-degree polynomials bloated by intervals for efficiency. Subdivision strategies limit scalability in long-time simulations.
Floating-Point Error Verification
Estimating round-off errors in Taylor expansions demands symbolic methods to avoid pessimistic bounds. Solovyev et al. (2018) use symbolic Taylor expansions for rigorous bounds. Integrating these with verified solvers poses ongoing challenges.
Essential Papers
Model completeness results for expansions of the ordered field of real numbers by restricted Pfaffian functions and the exponential function
Andrew O.M. Wilkie · 1996 · Journal of the American Mathematical Society · 411 citations
Taylor Model Flowpipe Construction for Non-linear Hybrid Systems
Xin Chen, Erika Ábrahám, Sriram Sankaranarayanan · 2012 · 194 citations
We propose an approach for verifying non-linear hybrid systems using higher-order Taylor models that are a combination of bounded degree polynomials over the initial conditions and time, bloated by...
On Taylor Model Based Integration of ODEs
Markus Neher, Kenneth R. Jackson, Nedialko S. Nedialkov · 2007 · SIAM Journal on Numerical Analysis · 122 citations
Interval methods for verified integration of initial value problems (IVPs) for ODEs have been used for more than 40 years. For many classes of IVPs, these methods are able to compute guaranteed err...
Rigorous Estimation of Floating-Point Round-Off Errors with Symbolic Taylor Expansions
Alexey Solovyev, Marek Baranowski, Ian Briggs et al. · 2018 · ACM Transactions on Programming Languages and Systems · 91 citations
Rigorous estimation of maximum floating-point round-off errors is an important capability central to many formal verification tools. Unfortunately, available techniques for this task often provide ...
Convex/concave relaxations of parametric ODEs using Taylor models
Ali M. Sahlodin, Benoît Chachuat · 2011 · Computers & Chemical Engineering · 60 citations
Verified Reachability Analysis of Continuous Systems
Fabian Immler · 2015 · Lecture notes in computer science · 59 citations
Real root isolation for exact and approximate polynomials using descartes' rule of signs
Arno Eigenwillig · 2008 · SciDok (Saarland University and State Library) · 55 citations
Collins und Akritas (1976) have described the Descartes method for isolating the real roots of an integer polynomial in one variable. This method recursively subdivides an initial interval until De...
Reading Guide
Foundational Papers
Start with Neher et al. (2007) for core ODE integration theory (122 citations), then Chen et al. (2012) for hybrid extensions (194 citations), as they establish enclosure tightness basics.
Recent Advances
Study Solovyev et al. (2018) for floating-point verification (91 citations) and Yi et al. (2019) for numerical library repairs using Taylor-inspired bounds.
Core Methods
Core techniques: polynomial expansion with remainder intervals (Neher et al., 2007), flowpipe construction via bloating (Chen et al., 2012), symbolic expansions for errors (Solovyev et al., 2018).
How PapersFlow Helps You Research Taylor Models for Verified Computing
Discover & Search
Research Agent uses searchPapers and citationGraph to map Taylor model literature from Chen et al. (2012, 194 citations) to Neher et al. (2007), then findSimilarPapers for ODE applications. exaSearch uncovers related verified computing works beyond OpenAlex.
Analyze & Verify
Analysis Agent applies readPaperContent on Chen et al. (2012) to extract flowpipe algorithms, verifyResponse with CoVe for enclosure tightness claims, and runPythonAnalysis to simulate Taylor model remainders with NumPy intervals. GRADE scores evidence on wrapping reduction (A-grade for Neher et al., 2007).
Synthesize & Write
Synthesis Agent detects gaps in high-dimensional Taylor models, flags contradictions between Solovyev et al. (2018) and traditional intervals. Writing Agent uses latexEditText for enclosure proofs, latexSyncCitations for 10+ papers, latexCompile for reports, and exportMermaid for flowpipe diagrams.
Use Cases
"Implement Taylor model integration for Lotka-Volterra ODE with verified bounds."
Research Agent → searchPapers 'Taylor ODE' → Analysis Agent → runPythonAnalysis (NumPy interval solver on Neher et al. 2007 method) → verified enclosure plot and error bounds.
"Write LaTeX section on Taylor models for hybrid system reachability."
Synthesis Agent → gap detection (Chen et al. 2012) → Writing Agent → latexEditText (add proofs) → latexSyncCitations (10 papers) → latexCompile → camera-ready PDF with diagrams.
"Find GitHub repos with Taylor model verified computing code."
Research Agent → citationGraph (Solovyev et al. 2018) → Code Discovery → paperExtractUrls → paperFindGithubRepo → githubRepoInspect → runnable NumPy/interval arithmetic implementations.
Automated Workflows
Deep Research workflow scans 50+ Taylor model papers via searchPapers → citationGraph, producing structured reports on ODE vs. hybrid applications with GRADE scores. DeepScan applies 7-step analysis: readPaperContent (Neher et al., 2007) → runPythonAnalysis verification → CoVe checkpoints. Theorizer generates new subdivision heuristics from Chen et al. (2012) flowpipes and Sahlodin (2011) relaxations.
Frequently Asked Questions
What defines a Taylor model?
A Taylor model is a pair of a multivariate Taylor polynomial and an interval remainder that encloses the function range while tracking dependencies to reduce wrapping (Neher et al., 2007).
What are core methods in Taylor models?
Methods include Taylor series expansion with interval bloating for remainders, adaptive subdivision for range bounding, and convex/concave relaxations for optimization (Chen et al., 2012; Sahlodin and Chachuat, 2011).
What are key papers on Taylor models?
Foundational works: Neher et al. (2007, 122 citations) on ODE integration; Chen et al. (2012, 194 citations) on hybrid flowpipes; Solovyev et al. (2018, 91 citations) on floating-point errors.
What open problems exist?
Challenges include scalable high-order models for long-time hybrid simulations and tight error bounds in high dimensions beyond current subdivision techniques (Chen et al., 2012).
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Part of the Numerical Methods and Algorithms Research Guide