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Universal approximations of quasilinear PDEs by finite distinguishable particle systems

Thierry Paul, Emmanuel Trélat

TL;DR

The paper establishes a constructive link between quasilinear PDEs and finite particle systems by showing that sufficiently regular PDE solutions can be approximated, with error decaying like $1/\log N$, by carefully designed $N$-particle dynamics derived from mollified operators. The method proceeds in two steps: first approximate the PDE operator with a bounded, mollified operator $A_\varepsilon$ and obtain a correspondingly mollified solution $y_\varepsilon$, then discretize via tagged partitions to build a particle system $y_\varepsilon^N$ that converges to $y_\varepsilon$ as $N\to\infty$; combining these steps yields quantitative convergence rates for $y_\varepsilon^N$ to $y$. The authors provide explicit particle constructions for a broad class of quasilinear PDEs, derive convergence in $L^2$ (and in $L^\infty$ under suitable embeddings), and present concrete examples (transport, Burgers, KdV). They also connect the analysis to graph-limit ideas and discuss extensions such as the backward heat equation and variational inequalities, offering a versatile framework for linking continuum PDEs to finite, interpretable particle systems with rigorous error control.

Abstract

In this paper, we prove that sufficiently regular solutions of any quasilinear PDE can be approximated by solutions of systems of N distinguishable particles, to within 1/ ln(N ). This intruiguing result is related to recent developments in graph limit theory.

Universal approximations of quasilinear PDEs by finite distinguishable particle systems

TL;DR

The paper establishes a constructive link between quasilinear PDEs and finite particle systems by showing that sufficiently regular PDE solutions can be approximated, with error decaying like , by carefully designed -particle dynamics derived from mollified operators. The method proceeds in two steps: first approximate the PDE operator with a bounded, mollified operator and obtain a correspondingly mollified solution , then discretize via tagged partitions to build a particle system that converges to as ; combining these steps yields quantitative convergence rates for to . The authors provide explicit particle constructions for a broad class of quasilinear PDEs, derive convergence in (and in under suitable embeddings), and present concrete examples (transport, Burgers, KdV). They also connect the analysis to graph-limit ideas and discuss extensions such as the backward heat equation and variational inequalities, offering a versatile framework for linking continuum PDEs to finite, interpretable particle systems with rigorous error control.

Abstract

In this paper, we prove that sufficiently regular solutions of any quasilinear PDE can be approximated by solutions of systems of N distinguishable particles, to within 1/ ln(N ). This intruiguing result is related to recent developments in graph limit theory.
Paper Structure (33 sections, 14 theorems, 93 equations)

This paper contains 33 sections, 14 theorems, 93 equations.

Table of Contents

  1. Introduction
  2. Setting
  3. Some tools and notations
  4. Tagged partitions, and Riemann sum theorem.
  5. Functional spaces.
  6. Finite particle approximations of quasilinear evolution systems
  7. Abstract quasilinear evolution systems
  8. A general approximation result
  9. Finite particle approximation system
  10. Main result
  11. Application to quasilinear PDEs
  12. A general class of quasilinear PDEs
  13. Main result
  14. Explicit particle approximation.
  15. Proof of Theorem \ref{['thm_approx_PDE_particle']}
  16. ...and 18 more sections

Key Result

Proposition 1

Let $y^0\in Z$. Under Assumptions H_Z to H_f_Lip, there exists a unique solution $y(\cdot)\in\mathscr{C}^0([0,T'],Z)\cap\mathscr{C}^1([0,T'],X)$ of abstract_quasilinear such that $y(0)=y^0$, for some $T'\in(0,T]$. Moreover, for every $t\in[0,T']$ one has $y(t)\in B_Z(y^0,r)$ and

Theorems & Definitions (34)

  • Proposition 1
  • Remark 1
  • proof
  • Remark 2
  • Proposition 2
  • proof
  • Proposition 3
  • proof
  • Theorem 1
  • proof
  • ...and 24 more