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Phase-IDENT: Identification of Two-phase PDEs with Uncertainty Quantification

Edward L. Yang, Roy Y. He

TL;DR

Phase-IDENT tackles the problem of discovering PDEs when a single spatial-temporal domain is governed by two different dynamics separated by an unknown phase boundary. It combines a patch-based PDE discovery approach with a change-point framework to locate the boundary and to quantify uncertainty, yielding phase-specific PDEs and a continuous boundary representation with confidence bands. The method uses Robust-IDENT for local model discovery, single-step evolution errors to signal the boundary, and Wasserstein barycenters to fuse boundary information, supported by convexification and spline-based boundary representation. Numerical experiments across noise levels and boundary slopes demonstrate robust PDE identification and accurate boundary localization, highlighting Phase-IDENT's applicability to multimode dynamics in fluids and materials.

Abstract

We propose a novel method, Phase-IDENT, for identifying partial differential equations (PDEs) from noisy observations of dynamical systems that exhibit phase transitions. Such phenomena are prevalent in fluid dynamics and materials science, where they can be modeled mathematically as functions satisfying different PDEs within distinct regions separated by phase boundaries. Our approach simultaneously identifies the underlying PDEs in each regime and accurately reconstructs the phase boundaries. Furthermore, by incorporating change point detection techniques, we provide uncertainty quantification for the detected boundaries, enhancing the interpretability and robustness of our method. We conduct numerical experiments on a variety of two-phase PDE systems under different noise levels, and the results demonstrate the effectiveness of the proposed approach.

Phase-IDENT: Identification of Two-phase PDEs with Uncertainty Quantification

TL;DR

Phase-IDENT tackles the problem of discovering PDEs when a single spatial-temporal domain is governed by two different dynamics separated by an unknown phase boundary. It combines a patch-based PDE discovery approach with a change-point framework to locate the boundary and to quantify uncertainty, yielding phase-specific PDEs and a continuous boundary representation with confidence bands. The method uses Robust-IDENT for local model discovery, single-step evolution errors to signal the boundary, and Wasserstein barycenters to fuse boundary information, supported by convexification and spline-based boundary representation. Numerical experiments across noise levels and boundary slopes demonstrate robust PDE identification and accurate boundary localization, highlighting Phase-IDENT's applicability to multimode dynamics in fluids and materials.

Abstract

We propose a novel method, Phase-IDENT, for identifying partial differential equations (PDEs) from noisy observations of dynamical systems that exhibit phase transitions. Such phenomena are prevalent in fluid dynamics and materials science, where they can be modeled mathematically as functions satisfying different PDEs within distinct regions separated by phase boundaries. Our approach simultaneously identifies the underlying PDEs in each regime and accurately reconstructs the phase boundaries. Furthermore, by incorporating change point detection techniques, we provide uncertainty quantification for the detected boundaries, enhancing the interpretability and robustness of our method. We conduct numerical experiments on a variety of two-phase PDE systems under different noise levels, and the results demonstrate the effectiveness of the proposed approach.
Paper Structure (22 sections, 1 theorem, 76 equations, 16 figures, 2 tables, 1 algorithm)

This paper contains 22 sections, 1 theorem, 76 equations, 16 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Proposed Method: Phase-IDENT
  3. Overview of Phase-IDENT
  4. Phase detection and initial cover construction
  5. PDE identification in phase domains
  6. Local evolution error across phase boundary
  7. Change point detection and uncertainty quantification
  8. Implementation Details
  9. Construction of the initial cover
  10. Construction of $\widetilde{\mathcal{C}}$ via convexification
  11. Continuous representation for the phase boundary estimation
  12. Numerical Experiments
  13. General performances
  14. Performances under different noise levels
  15. Effects of the phase boundary slope
  16. ...and 7 more sections

Key Result

Proposition 2.1

For $x \in \mathcal{D}$, if $\bm{U}_m(x) \subseteq \Omega_1$ and $\tau_m < \gamma(x) < \tau_{m+1}$, let and Then

Figures (16)

  • Figure 1: Noisy observations of a solution governed by two distinct PDEs in separate domains. (a) A simulated solution governed by the transport equation followed by the viscous Burgers equation, with $10\%$ additive noise. (b) The numerical solution of a globally identified PDE using the data from (a). (c) The green dashed line indicates the ground truth location of the phase boundary. The white solid line shows the phase boundary estimated by Phase-IDENT, while the white dashed lines represent the corresponding confidence intervals at an $80\%$ confidence level (Section \ref{['boundary estimation']}).
  • Figure 2: Schematic diagram of the proposed Phase-IDENT workflow. Phase-IDENT begins by covering the time–space domain $\Omega$ with patches. For each patch, a PDE is identified and its fit is assessed via the cross‑validation estimation error (CEE) defined in \ref{['CEE formu']}. If the distribution of high‑CEE patches is scattered (Section \ref{['patch detection']}), only a single phase is inferred, and a unique PDE is identified using the entire dataset. Otherwise, the high‑CEE patches are used to construct an open set $\widetilde{\mathcal{C}}$. PDEs are then identified separately within each connected component of $\Omega \setminus \widetilde{\mathcal{C}}$ (Section \ref{['sec_PDE_ident']}), and their numerical evolution errors (Section \ref{['evolution section']}) inside $\widetilde{\mathcal{C}}$ are employed to locate the phase boundary via change point detection (Section \ref{['boundary estimation']}).
  • Figure 3: Comparison of the distributions of CEE values for PDEs identified from patches that contain (orange boxes) and do not contain (blue boxes) phase boundaries. The underlying two-phase PDEs are: (a) Transport $\to$ Viscous Burgers (T$\to$VB); (b) KdV $\to$ Burgers (KdV$\to$B); and (c) Burgers $\to$ Transport (B$\to$T). Across a range of noise-to-signal ratios (NSRs) in the observed data, patches with high CEE values serve as consistent indicators of phase boundary presence.
  • Figure 4: Comparison of the distributions of patches with high CEE values \ref{['CEE formu']} when locally identifying PDEs from data governed by single equations: (a) Transport; (b) KdV; and (c) Burgers; and from data governed by two-phase equations: (d) Transport $\to$ Viscous Burgers (T$\to$VB); (e) KdV $\to$ Burgers (KdV$\to$B); and (f) Burgers $\to$ Transport (B$\to$T). Patches with high CEE are scattered in (a)–(c), whereas they concentrate around the respective phase boundaries in (d)–(f). Using the proposed indicator $r_{\beta}$\ref{['ra']}, the values for (a), (b), and (c) are 0.28, 0.65, and 1.00, respectively; while for (d), (e), and (f), they are 38.02, 16.50, and 8.92, respectively. Throughout this experiment, $\beta = 10$.
  • Figure 5: (a) Noisy observations of a trajectory governed first by the KdV equation and then by the Burgers equation. The green dashed line indicates the true phase boundary. The PDEs identified on both sides are numerically evolved along horizontal slices (blue dashed lines). (b) Simulation error sequences $\mathbf{e}^{\ell}(x)$ (blue) and $\mathbf{e}^{r}(x)$ (orange) defined in \ref{['eq_sequences']}. (c) Density functions $p^\ell_{x}$ (blue) and $p^r_{x}$ (red) defined in \ref{['pdelta']}, and their Wasserstein barycenter $p_{x}$ (green) defined in \ref{['ot']}. The vertical brown dashed line in (c) marks the ground truth location of the boundary point. Here $x = -1.885$ and the data contain $2\%$ additive noise.
  • ...and 11 more figures

Theorems & Definitions (5)

  • Remark 2.1
  • Proposition 2.1
  • proof
  • Remark 2.2
  • proof