arXiv subjects

Marcio Gameiro

Publications and source records attributed to Marcio Gameiro.

At least 19 recordsLinked to original sources

arXiv2026

A Lean~4 Framework for the Radii Polynomial Method

Understanding nonlinear dynamics is central to many fields of study, yet most nonlinear systems are beyond the reach of closed-form solutions. Numerical simulation is then used to reveal dynamical structures before their existence can be established. This alone, however, leaves the validity of the computed solution unanswered. Rigorous numerical methods address this gap by combining computation with explicit analytic estimates. Among them, the radii polynomial approach, a Newton--Kantorovich type argument, turns a numerical approximation into a proof of existence and uniqueness of a true solution near that approximation. In previous practice, the verification of these inequalities relies on interval-arithmetic libraries and hand-derived algebraic bounds. In this paper, we present a framework for the radii polynomial method in the Lean~4, which turns numerical certificates into fully type-checked theorems about the true solution. A central challenge in this process is to manage the complexity of the formalization. Our approach begins with a family of problems and their common structure. The spaces in which the analysis takes place are developed through universal properties in suitable categories, with extension properties that characterize maps by their action on generators. These properties guide the design of a reusable interface such that problem-specific arguments concentrate on the numerical data and the algebraic bounds that distinguish one equation from another. We formalize a general radii polynomial theorem, develop weighted coefficient algebras and reusable operator estimates, connect finite computational checks to the hypotheses of the theorem, and establish the realization theorems that transport certified coefficient space zeros to analytic solutions of the original equations.

math.DS↗

arXiv2026

Characterizing High-dimensional Dynamics by Combinatorial-Topological Methods on a Latent Space

Combinatorial-topological methods for characterizing dynamics are rigorous, generalizable, computable, and they only require approximations, but the dimension of the phase space is a computational bottleneck to their wider application. Motivated by the growing number of machine learning techniques for obtaining lower-dimensional latent representations of dynamics, we present an initial study of combinatorial-topological techniques in the dimensionality reduction setting. We establish bounds under which an algebraic structure that organizes dynamics can be lifted from the latent space to the original system. As a corollary, one can conclude the existence of attractors within certain regions of the original phase space. The hypothesis of these results is expressed in terms of an approximate semiconjugacy between the original and latent dynamics. To demonstrate the ideas, we combine autoencoder-based models with Conley-Morse graph computations for Leslie population models, a thirteen-dimensional Mediterranean red coral population model, and the Chafee--Infante equation. While the lift of the Conley index is still an open question, the examples recover the expected algebraic topological invariants in several settings.

math.DS↗

arXiv2026

Boolean models coarsely sample continuous dynamics of regulatory networks

Boolean models are widely used to characterize the dynamics of gene regulatory networks. However, their coarse state discretization limits their ability to capture complex continuous dynamics and continuous parameter dependencies. In this paper, we present a rigorous mathematical framework that embeds monotone Boolean models into a broader class of multilevel combinatorial models, which in turn embed into the Dynamic Signatures Generated by Regulatory Networks (DSGRN) methodology. We define the DSGRN parameter graph, which encodes the notion of parameter adjacency and is used to map Boolean functions to specific nodes within the DSGRN parameter space. We prove that these multilevel discrete update functions act as a multilevel refinement of monotone Boolean models. We demonstrate that purely Boolean models systematically underestimate network dynamics by missing crucial intermediate behaviors such as higher-order multistability and stable periodic orbits. We show that the DSGRN framework efficiently captures a strictly richer set of dynamics consistent with ordinary differential equations (ODEs), providing a mathematically rigorous and computationally viable bridge between discrete and continuous network modeling.

q-bio.MN↗

arXiv2025

Rigorously Characterizing Dynamics with Machine Learning

The identification of dynamics from time series data is a problem of general interest. It is well established that dynamics on the level of invariant sets, the primary objects of interest in the classical theory of dynamical systems, is not computable. We recall a coarser characterization of dynamics based on order theory and algebraic topology and prove that this characterization can be identified using approximations.

math.DS↗

arXiv2025

Data-driven Identification of Attractors Using Machine Learning

In this paper we explore challenges in developing a topological framework in which machine learning can be used to robustly characterize global dynamics. Specifically, we focus on learning a useful discretization of the phase space of a flow on compact, hyperrectangle in $\mathbb{R}^n$ from a neural network trained on labeled orbit data. A characterization of the structure of the global dynamics is obtained from approximations of attracting neighborhoods provided by the phase space discretization. The perspective that motivates this work is based on Conley's topological approach to dynamics, which provides a means to evaluate the efficacy and efficiency of our approach.

math.DS↗

arXiv2024

Global Dynamics of Ordinary Differential Equations: Wall Labelings, Conley Complexes, and Ramp Systems

We introduce a combinatorial topological framework for characterizing the global dynamics of ordinary differential equations (ODEs). The approach is motivated by the study of gene regulatory networks, which are often modeled by ODEs that are not explicitly derived from first principles. The proposed method involves constructing a combinatorial model from a set of parameters and then embedding the model into a continuous setting in such a way that the algebraic topological invariants are preserved. In this manuscript, we build upon the software Dynamic Signatures Generated by Regulatory Networks (DSGRN), a software package that is used to explore the dynamics generated by a regulatory network. By extending its functionalities, we deduce the global dynamical information of the ODE and extract information regarding equilibria, periodic orbits, connecting orbits and bifurcations. We validate our results through algebraic topological tools and analytical bounds, and the effectiveness of this framework is demonstrated through several examples and possible future directions.

math.DS↗

arXiv2023

Towards computational Morse-Floer homology: forcing results for connecting orbits by computing relative indices of critical points

To make progress towards better computability of Morse-Floer homology, and thus enhance the applicability of Floer theory, it is essential to have tools to determine the relative index of equilibria. Since even the existence of nontrivial stationary points is often difficult to accomplish, extracting their index information is usually out of reach. In this paper we establish a computer-assisted proof approach to determining relative indices of stationary states. We introduce the general framework and then focus on three example problems described by partial differential equations to show how these ideas work in practice. Based on a rigorous implementation, with accompanying code made available, we determine the relative indices of many stationary points. Moreover, we show how forcing results can be then used to prove theorems about connecting orbits and traveling waves in partial differential equations.

math.DS↗

arXiv2023

Identifying Nonlinear Dynamics with High Confidence from Sparse Data

We introduce a novel procedure that, given sparse data generated from a stationary deterministic nonlinear dynamical system, can characterize specific local and/or global dynamic behavior with rigorous probability guarantees. More precisely, the sparse data is used to construct a statistical surrogate model based on a Gaussian process (GP). The dynamics of the surrogate model is interrogated using combinatorial methods and characterized using algebraic topological invariants (Conley index). The GP predictive distribution provides a lower bound on the confidence that these topological invariants, and hence the characterized dynamics, apply to the unknown dynamical system (a sample path of the GP). The focus of this paper is on explaining the ideas, thus we restrict our examples to one-dimensional systems and show how to capture the existence of fixed points, periodic orbits, connecting orbits, bistability, and chaotic dynamics.

math.DS↗

arXiv2023

Inferring Long-term Dynamics of Ecological Communities Using Combinatorics

In an increasingly changing world, predicting the fate of species across the globe has become a major concern. Understanding how the population dynamics of various species and communities will unfold requires predictive tools that experimental data alone can not capture. Here, we introduce our combinatorial framework, Widespread Ecological Networks and their Dynamical Signatures (WENDyS) which, using data on the relative strengths of interactions and growth rates within a community of species predicts all possible long-term outcomes of the community. To this end, WENDyS partitions the multidimensional parameter space (formed by the strengths of interactions and growth rates) into a finite number of regions, each corresponding to a unique set of coarse population dynamics. Thus, WENDyS ultimately creates a library of all possible outcomes for the community. On the one hand, our framework avoids the typical ``parameter sweeps'' that have become ubiquitous across other forms of mathematical modeling, which can be computationally expensive for ecologically realistic models and examples. On the other hand, WENDyS opens the opportunity for interdisciplinary teams to use standard experimental data (i.e., strengths of interactions and growth rates) to filter down the possible end states of a community. To demonstrate the latter, here we present a case study from the Indonesian Coral Reef. We analyze how different interactions between anemone and anemonefish species lead to alternative stable states for the coral reef community, and how competition can increase the chance of exclusion for one or more species. WENDyS, thus, can be used to anticipate ecological outcomes and test the effectiveness of management (e.g., conservation) strategies.

q-bio.PE↗

arXiv2023

A framework for rigorous computational methods using Haar wavelets for differential equations

This work presents a framework for a-posteriori error-estimating algorithms for differential equations which combines the radii polynomial approach with Haar wavelets. By using Haar wavelets, we obtain recursive structures for the matrix representations of the differential operators and quadratic nonlinearities, which can be exploited for the radii polynomial method in order to get error estimates in the $L^2$ sense. This allows the method to be applicable when the system or solution is not continuous, which is a limitation of other radii-polynomial-based methods. Numerical examples show how the method is implemented in practice.

math.NA↗

arXiv2023

A yeast cell cycle pulse generator model shows consistency with multiple oscillatory and checkpoint mutant datasets

The regulatory mechanisms driving progression of the yeast cell cycle appears to be comprised of an interacting network of transcription factors (TFs), cyclin-dependent kinases (CDK) and ubiquitin ligases. From a systems perspective the controlling regulatory network must produce robust periodic behavior during proliferative phases, but have the capability to halt the cycle when unfavorable conditions trigger a checkpoint arrest. How the individual components of the network contribute to these dynamical phenotypes remains an open question. Here we evaluate the capability of a simplified network model hypothesized to contain key elements of the regulation of cell-cycle progression to reproduce observed transcriptomic behaviors. We match time-series data from both cycling and checkpoint arrested cells to the predictions of a relatively simple cell-cycle network model using an asynchronous multi-level Boolean approach. We show that this single network model, despite its simplicity, is capable of exhibiting dynamical behavior similar to the datasets in most cases, and where it does not, we identified hypotheses that suggest missing components of the network.

math.DS↗

arXiv2022

Data-Efficient Characterization of the Global Dynamics of Robot Controllers with Confidence Guarantees

This paper proposes an integration of surrogate modeling and topology to significantly reduce the amount of data required to describe the underlying global dynamics of robot controllers, including closed-box ones. A Gaussian Process (GP), trained with randomized short trajectories over the state-space, acts as a surrogate model for the underlying dynamical system. Then, a combinatorial representation is built and used to describe the dynamics in the form of a directed acyclic graph, known as {\it Morse graph}. The Morse graph is able to describe the system's attractors and their corresponding regions of attraction (\roa). Furthermore, a pointwise confidence level of the global dynamics estimation over the entire state space is provided. In contrast to alternatives, the framework does not require estimation of Lyapunov functions, alleviating the need for high prediction accuracy of the GP. The framework is suitable for data-driven controllers that do not expose an analytical model as long as Lipschitz-continuity is satisfied. The method is compared against established analytical and recent machine learning alternatives for estimating \roa s, outperforming them in data efficiency without sacrificing accuracy. Link to code: https://go.rutgers.edu/49hy35en

cs.RO↗

arXiv2022

Morse Graphs: Topological Tools for Analyzing the Global Dynamics of Robot Controllers

Understanding the global dynamics of a robot controller, such as identifying attractors and their regions of attraction (RoA), is important for safe deployment and synthesizing more effective hybrid controllers. This paper proposes a topological framework to analyze the global dynamics of robot controllers, even data-driven ones, in an effective and explainable way. It builds a combinatorial representation representing the underlying system's state space and non-linear dynamics, which is summarized in a directed acyclic graph, the Morse graph. The approach only probes the dynamics locally by forward propagating short trajectories over a state-space discretization, which needs to be a Lipschitz-continuous function. The framework is evaluated given either numerical or data-driven controllers for classical robotic benchmarks. It is compared against established analytical and recent machine learning alternatives for estimating the RoAs of such controllers. It is shown to outperform them in accuracy and efficiency. It also provides deeper insights as it describes the global dynamics up to the discretization's resolution. This allows to use the Morse graph to identify how to synthesize controllers to form improved hybrid solutions or how to identify the physical limitations of a robotic system.

cs.RO↗

arXiv2022

Rigorous Enclosures of Solutions of Neumann Boundary Value Problems

This paper is dedicated to the problem of isolating and validating zeros of non-linear two point boundary value problems. We present a method for such purpose based on the Newton-Kantorovich Theorem to rigorously enclose isolated zeros of two point boundary value problem with Neumann boundary conditions.

math.AP↗

arXiv2022

Extending combinatorial regulatory network modeling to include activity control and decay modulation

Understanding how the structure of within-system interactions affects the dynamics of the system is important in many areas of science. We extend a network dynamics modeling platform DSGRN, which combinatorializes both dynamics and parameter space to construct finite but accurate summaries of network dynamics, to new types of interactions. While the standard DSGRN assumes that each network edge controls the rate of abundance of the target node, the new edges may control either activity level or a decay rate of its target. While motivated by processes of post-transcriptional modification and ubiquitination in systems biology, our extension is applicable to the dynamics of any signed directed network.

math.DS↗

arXiv2021

Extracting Global Dynamics of Loss Landscape in Deep Learning Models

Deep learning models evolve through training to learn the manifold in which the data exists to satisfy an objective. It is well known that evolution leads to different final states which produce inconsistent predictions of the same test data points. This calls for techniques to be able to empirically quantify the difference in the trajectories and highlight problematic regions. While much focus is placed on discovering what models learn, the question of how a model learns is less studied beyond theoretical landscape characterizations and local geometric approximations near optimal conditions. Here, we present a toolkit for the Dynamical Organization Of Deep Learning Loss Landscapes, or DOODL3. DOODL3 formulates the training of neural networks as a dynamical system, analyzes the learning process, and presents an interpretable global view of trajectories in the loss landscape. Our approach uses the coarseness of topology to capture the granularity of geometry to mitigate against states of instability or elongated training. Overall, our analysis presents an empirical framework to extract the global dynamics of a model and to use that information to guide the training of neural networks.

math.DS↗

arXiv2021

Rational design of complex phenotype via network models

We demonstrate a modeling and computational framework that allows for rapid screening of thousands of potential network designs for particular dynamic behavior. To illustrate this capability we consider the problem of hysteresis, a prerequisite for construction of robust bistable switches and hence a cornerstone for construction of more complex synthetic circuits. We evaluate and rank most three node networks according to their ability to robustly exhibit hysteresis where robustness is measured with respect to parameters over multiple dynamic phenotypes. Focusing on the highest ranked networks, we demonstrate how additional robustness and design constraints can be applied. We compare our results to more traditional methods based on specific parameterization of ordinary differential equation models and demonstrate a strong qualitative match at a small fraction of the computational cost.

math.DS↗

arXiv2020

On the feasibility and convergence of the inexact Newton method under minor conditions on the error terms

In this paper we introduce a semi-local theorem for the feasibility and convergence of the inexact Newton method, regarding the sequence $x_{k+1} = x_k - Df(x_k)^{-1}f(x_k) + r_k$, where $r_k$ represents the error in each step. Unlike the previous results of this type in the literature, we prove the feasibility of the inexact Newton method under the minor hypothesis that the error $r_k$ is bounded by a small constant to be computed, and moreover we prove results concerning the convergence of the sequence $x_k$ to the solution under this hypothesis. Moreover, we show how to apply this this method to compute rigorously zeros for two-point boundary value problems of Neumann type. Finally, we apply it to a version of the Cahn-Hilliard equation.

math.AP↗