Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The Evolution of Computer-Assisted Proof In Analysis

Marek Rychlik

Abstract

The intersection of numerical analysis and machine learning, particularly in the domain of Neural ODEs and Physics-Informed Neural Networks (PINNs), relies heavily on discrete approximations of continuous flows. However, in stiff systems, explicit discretization schemes can induce topological bifurcations, creating spurious attractors that do not exist in the underlying continuous dynamics. In this paper, we analyze a stiff 2D nonlinear system integrated via Heun's method, demonstrating how the discrete map undergoes a numerical bifurcation that renders the true equilibrium repelling along an invariant manifold. Adopting a literate programming paradigm where "the paper is the proof," we embed a Computer-Assisted Proof (CAP) directly within the manuscript. Utilizing rigorous complex interval arithmetic and a dimensionality-reducing "Snap-to-Axis" projection, we mathematically verify that a neighborhood of trajectories is permanently captured by a spurious period-4 sink. This work highlights the critical need for formal safety certifications in learned dynamical models and provides a framework for verifying their stability.

The Evolution of Computer-Assisted Proof In Analysis

Abstract

The intersection of numerical analysis and machine learning, particularly in the domain of Neural ODEs and Physics-Informed Neural Networks (PINNs), relies heavily on discrete approximations of continuous flows. However, in stiff systems, explicit discretization schemes can induce topological bifurcations, creating spurious attractors that do not exist in the underlying continuous dynamics. In this paper, we analyze a stiff 2D nonlinear system integrated via Heun's method, demonstrating how the discrete map undergoes a numerical bifurcation that renders the true equilibrium repelling along an invariant manifold. Adopting a literate programming paradigm where "the paper is the proof," we embed a Computer-Assisted Proof (CAP) directly within the manuscript. Utilizing rigorous complex interval arithmetic and a dimensionality-reducing "Snap-to-Axis" projection, we mathematically verify that a neighborhood of trajectories is permanently captured by a spurious period-4 sink. This work highlights the critical need for formal safety certifications in learned dynamical models and provides a framework for verifying their stability.
Paper Structure (9 sections, 3 theorems, 13 equations, 4 figures, 2 tables)

This paper contains 9 sections, 3 theorems, 13 equations, 4 figures, 2 tables.

Table of Contents

  1. Historical Background and the Evolution of Rigor
  2. The Lanford Milestone and the Lorenz Attractor
  3. Literate Programming and the Logic of Proof
  4. From Specific Lemmas to Systematic Integration
  5. The Non-Invertible Challenge and Invariant Manifolds
  6. The Modern Paradigm: Symmetry and Projection
  7. The Motivating Example: Numerical Bifurcations in Stiff Systems
  8. Stiffness as a Bifurcation Parameter and Universality
  9. Discussion and Concluding Remarks

Key Result

Lemma 1

If $F$ were a diffeomorphism, then the global stable manifold $W^s(0)$ would be an embedded 1D manifold. However, due to $F$ not being 1:1, $W^s(0)$ is more complex, with multiple branches. The point $(0,8.31177)$ represents a non-local pre-image of the origin, marking where a second branch of the m

Figures (4)

  • Figure 1: Phase diagram of the system \ref{['eqn:stiff-study']}. The numerical trajectory is marked in red.
  • Figure 2: Graph of the 1D map $g(x)$ .
  • Figure 3: Convergence of the trajectory of the critical point to period 4 periodic orbit.
  • Figure 4: The basin of attraction of the period-4 periodic orbit. The periodic orbit is marked in yellow, with the initial condition highlighted in red.

Theorems & Definitions (10)

  • Example 1: Nonlinear System: Heun's Method
  • Definition 1: The Discrete Map
  • Lemma 1: The Folding Point
  • Remark 1: Stability Function and Linear Analysis
  • Theorem 1: Hyperbolicity and Stable Manifolds
  • Definition 2: Interval Arithmetic
  • Remark 2: On the software used
  • Remark 3: The Role of the "Snap-to-Axis" Strategy
  • Theorem 2: Rigorous Absorption and Invariance
  • Remark 4: On decimal vs. machine numbers