arXiv subjects

Maximilian Topel

Publications and source records attributed to Maximilian Topel.

4 recordsLinked to original sources

arXiv2026

Phase spaces of phase spaces: A method of constructing generalizable and topologically informed bases from time series demonstrated by reconstruction of the geometry of the macroeconomy

Laws from physics to social science rest on a small, enumerable number of factors explaining variation in the systems they describe. While the orthodoxy of economics, physics and chemistry relies on a small number of analytically interpretable laws, large data models pursuing universality throw in all available data. Both fail in their extremes through the same flaw, an incorrect basis: too few orthogonal variables to describe system variation, or an explosion of dimensionality around fundamentally low dimensional but unknown dynamics. We propose and validate a framework that extracts an interpretable, mathematically sound, low dimensional and dynamically informed basis directly from arbitrary timeseries. Combining Takens' Delay Embedding Theorem, dimensionality reduction and optimal transport, we produce a phase space of phase spaces (PSoPS) on which all interrelated dynamical series exist: barycenters formed from many phase spaces, together with the families of transport plans between individual phase spaces and the barycenter, yield a basis describing the relational geometry of each timeseries and the topology of economic relationships. The macroeconomic PSoPS built from 281,536 series is a single connected object of effective dimension 7.3 +/- 0.2, into which the canonical laws (Phillips, Okun, Solow) embed as roughly 2-dimensional looped sub-attractors. The coupling between concept barycenters is directed, with the price of money and productivity as net sources and credit and output as sinks, recovering monetary transmission from geometry alone. IAAFT phase-randomized surrogates destroy 36-43% of the dimension across the three laws and 38% on the full corpus, so the basis is irreducibly nonlinear. We close with proposed extensions to fields whose dynamics share attractor geometries and to systems that are intrinsically low dimensional yet hard to navigate in raw coordinates.

physics.soc-ph↗

arXiv2026

Data-driven reconstruction of dynamical systems using Takens' Theorem, manifold learning, and universal function approximators

Embedding theorems can be used to provide theoretical guarantees about the relation between low-dimensional observations of a system and its full-dimensional state and dynamics. Such theorems do not, however, provide guidance on observable choice, embedding construction, or methodologies to learn the mapping between the embedding and full-dimensional state. In this work, we develop an algorithmic framework, TAkens Reconstruction (TAR), to analyze and reconstruct arbitrary dynamical systems from low-dimensional time series using an integration of Takens' Delay Embedding Theorem, manifold learning techniques, and universal function approximators. We validate TAR in applications to a variety of simulated and observed dynamical systems and use it to investigate how delay vector structure impacts reconstruction accuracy. In an ecological system, we show that simple predator-prey dynamics can be reconstructed with observations taken over a wide variety of embedding time scales. In molecular dynamics simulations of the protein Villin, we demonstrate how including multiple time delays of the same observable series can be used to improve reconstruction of systems with multiple characteristic time scales. In the trade record of Vanguard S&P 500, we show how the approach exposes underlying dynamical phenomenologies in the data and accurate return predictions over short time horizons without access to full-dimensional market observations. We develop and release an open-source software package to enable the application of TAR to arbitrary dynamical systems.

physics.comp-ph↗

arXiv2026

Kolmogorov-Sinai entropies identify optimal observables for prediction and dynamics reconstruction in chaotic systems

Choosing the optimal observable to model dynamical systems for which we do not know the driving equations is nearly always an ad hoc art. Takens' Delay Embedding Theorem guarantees a diffeomorphism between delay-coordinate vectors built from generic scalar observables and the underlying invariant attractor, but is agnostic to optimal observable choice, and formal bounds on reconstruction quality across observables are not known. Here we prove that, under modest technical conditions, the Kolmogorov-Sinai entropy of an observable predicts its reconstruction error of the underlying dynamics in chaotic, ergodic systems. Using the Oseledets Multiplicative Ergodic Theorem, we show that the tangent bundles of reconstructed manifolds admit an invariant Oseledets filtration diffeomorphically related across admissible observables, with Lyapunov exponents controlling the propagation of perturbations. We bound reconstruction error by a quantity monotonically related to the sum of positive Lyapunov exponents and, by the Ruelle inequality, the Kolmogorov-Sinai entropy. We validate this empirically on the Lorenz-63 attractor, the Hastings-Powell food chain, and a tetracosane molecular-dynamics trajectory, recovering Spearman rank correlations between $h^{KS,UB}$ and reconstruction RMSE up to $\rho=+0.89$ ($p=5.5\times 10^{-8}$) for the realistic tetracosane case, sharpening to $\rho=+0.97$ under added measurement noise. This provides a rigorous foundation for observable selection in chaotic systems, applicable as an a priori data-selection criterion for any data-driven modeling pipeline.

physics.comp-ph↗

arXiv2020

Reconstruction of Protein Structures from Single-Molecule Time Series

Single-molecule experimental techniques track the real-time dynamics of molecules by recording a small number of experimental observables. Following these observables provides a coarse-grained, low-dimensional representation of the conformational dynamics but does not furnish an atomistic representation of the instantaneous molecular structure. Takens' Delay Embedding Theorem asserts that, under quite general conditions, these low-dimensional time series can contain sufficient information to reconstruct the full molecular configuration of the system up to an a priori unknown transformation. By combining Takens' Theorem with tools from statistical thermodynamics, manifold learning, artificial neural networks, and rigid graph theory, we establish an approach Single-molecule TAkens Reconstruction (STAR) to learn this transformation and reconstruct molecular configurations from time series in experimentally-measurable observables such as intramolecular distances accessible to single molecule Förster resonance energy transfer. We demonstrate the approach in applications to molecular dynamics simulations of a C24H50 polymer chain and the artificial mini-protein Chignolin. The trained models reconstruct molecular configurations from synthetic time series data in the head-to-tail molecular distances with atomistic root mean squared deviation accuracies better than 0.2 nm. This work demonstrates that it is possible to accurately reconstruct protein structures from time series in experimentally-measurable observables and establishes the theoretical and algorithmic foundations to do so in applications to real experimental data.

physics.comp-ph↗