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BlinDNO: A Distributional Neural Operator for Dynamical System Reconstruction from Time-Label-Free data

Zhijun Zeng, Junqing Chen, Zuoqiang Shi

TL;DR

This work tackles the challenge of reconstructing dynamical operators from unordered density observations collected without explicit time labels. It introduces BlinDNO, a permutation‑invariant distribution‑to‑function neural operator that combines a multiscale U‑Net imaging operator with an attention‑based mixer and a Fourier Neural Operator refinement to recover operator parameters governing stochastic and quantum dynamics. The approach is validated across 1D and 2D SDE/Schrödinger systems and a high‑dimensional 3D cryo‑EM–inspired protein folding scenario, consistently outperforming Neural Inverse Operator baselines in both parameter and density reconstruction. By enabling direct inversion from density distributions under arbitrary observation‑time distributions, BlinDNO offers a scalable, robust tool for inverse dynamical problems with wide implications for cryo‑EM, molecular dynamics, and PDE‑constrained learning.

Abstract

We study an inverse problem for stochastic and quantum dynamical systems in a time-label-free setting, where only unordered density snapshots sampled at unknown times drawn from an observation-time distribution are available. These observations induce a distribution over state densities, from which we seek to recover the parameters of the underlying evolution operator. We formulate this as learning a distribution-to-function neural operator and propose BlinDNO, a permutation-invariant architecture that integrates a multiscale U-Net encoder with an attention-based mixer. Numerical experiments on a wide range of stochastic and quantum systems, including a 3D protein-folding mechanism reconstruction problem in a cryo-EM setting, demonstrate that BlinDNO reliably recovers governing parameters and consistently outperforms existing neural inverse operator baselines.

BlinDNO: A Distributional Neural Operator for Dynamical System Reconstruction from Time-Label-Free data

TL;DR

This work tackles the challenge of reconstructing dynamical operators from unordered density observations collected without explicit time labels. It introduces BlinDNO, a permutation‑invariant distribution‑to‑function neural operator that combines a multiscale U‑Net imaging operator with an attention‑based mixer and a Fourier Neural Operator refinement to recover operator parameters governing stochastic and quantum dynamics. The approach is validated across 1D and 2D SDE/Schrödinger systems and a high‑dimensional 3D cryo‑EM–inspired protein folding scenario, consistently outperforming Neural Inverse Operator baselines in both parameter and density reconstruction. By enabling direct inversion from density distributions under arbitrary observation‑time distributions, BlinDNO offers a scalable, robust tool for inverse dynamical problems with wide implications for cryo‑EM, molecular dynamics, and PDE‑constrained learning.

Abstract

We study an inverse problem for stochastic and quantum dynamical systems in a time-label-free setting, where only unordered density snapshots sampled at unknown times drawn from an observation-time distribution are available. These observations induce a distribution over state densities, from which we seek to recover the parameters of the underlying evolution operator. We formulate this as learning a distribution-to-function neural operator and propose BlinDNO, a permutation-invariant architecture that integrates a multiscale U-Net encoder with an attention-based mixer. Numerical experiments on a wide range of stochastic and quantum systems, including a 3D protein-folding mechanism reconstruction problem in a cryo-EM setting, demonstrate that BlinDNO reliably recovers governing parameters and consistently outperforms existing neural inverse operator baselines.

Paper Structure

This paper contains 19 sections, 2 theorems, 75 equations, 6 figures, 1 table.

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. Problem I: SDE driven dynamic
  4. Problem II: Quantum-mechanical dynamic
  5. Method
  6. Neural Operators between Function Spaces
  7. Deep Operator Network (DeepONet)
  8. Spectral Neural Operator
  9. Neural Operator from distribution to functions
  10. BlinDNO
  11. U-Net as a Neural Operator
  12. Permutation-invariant UNet
  13. Numerical Experiments
  14. Example 1: 1D Fokker–Planck Equation
  15. Example 2: 1D Linear/Nonlinear Schrödinger Equation
  16. ...and 4 more sections

Key Result

Lemma 2.1

Suppose $X_t$ solves the SDE eqn:sde, then the probability density function $\rho(x,t)$ satisfies the following d-dimensional Fokker-Planck equation by the Itô integral where $t\in[0,T]\subset\mathbb{R}$, $\mu = \bigl[\mu_1(x,t),\,\mu_2(x,t),\,\dots,\,\mu_d(x,t)\bigr]^T,$ and the diffusion matrix $[D_{ij}]=[D_{ij}(x,t)]$ is given by

Figures (6)

  • Figure 1: (a) Schematic illustration of the time-Label-Free dynamical system reconstruction problem. (b)The architecture of the BlinDNO.
  • Figure 1: Inversion results for the one-dimensional Fokker–Planck equation (FPE) problem. (a) Reconstructed potential profile, $U(x)$. (b) Reconstructed diffusion term. (c) Comparison between the density $\rho(T)$ obtained from simulations of the reconstructed dynamical system and the ground truth. (d) Relative $L_{2}$ error at each time step between the simulated density and the ground truth. (e) Ground-truth density distribution, $\rho_{\mathrm{GT}}(x,t)$. (f)–(h) Pointwise errors in the simulated density function $\rho(x,t)$, obtained using the reconstructed potential and diffusion term with BlinDNO, NIO, and FNO–NIO.
  • Figure 2: Inversion results for the 1D linear/nonlinear Schrodinger equation problem. (a) Ground-truth density distribution of the 1D linear Schrodinger equation, $\rho_{\mathrm{GT}}(x,t)$. (b) Reconstructed potential of the 1D linear Schrodinger equation, $V(x)$. (c) Comparison between the density $\rho(T)$ obtained from simulations of the reconstructed dynamical system and the ground truth. (d) Relative $L_{2}$ error at each time step between the simulated density and the ground truth. (a) Ground-truth density distribution of the 1D Gross-Pitaevskii equation, $\rho_{\mathrm{GT}}(x,t)$. (b) Reconstructed potential of the 1D 1D Gross-Pitaevskii equation, $V(x)$. (c) Comparison between the density $\rho(T)$ obtained from simulations of the reconstructed dynamical system and the ground truth. (d) Relative $L_{2}$ error at each time step between the simulated density and the ground truth.
  • Figure 3: Inversion results for the 2D Fokker–Planck equation with a non-uniform diffusion term. (a) Temporal evolution of the ground-truth density, $\rho_{\mathrm{GT}}(x,t)$. (b) Relative $L_{2}$ error at each time step between the simulated density and the ground truth. (c) Ground-truth potential, $U(x)$. (d)–(f) Reconstructed potential profiles, $U(x)$, obtained using BlinDNO, NIO, and FNO–NIO. (g) Ground-truth diffusion term, $D(x)$. (h)–(j) Reconstructed diffusion terms, $D(x)$, obtained using BlinDNO, NIO, and FNO–NIO.
  • Figure 4: Inversion results for the 2D Fokker–Planck equation with a non-conservative force field. (a) Temporal evolution of the ground-truth density, $\rho_{\mathrm{GT}}(x,t)$. (b) Relative $L_{2}$ error at each time step between the simulated density and the ground truth. (c) $x$-component of the ground-truth force field, $F_x(x)$. (d)–(f) Error of the reconstructed $x$-component of the force field, $F_x(x)$, obtained using BlinDNO, NIO, and FNO–NIO. (g) $y$-component of the ground-truth force field, $F_y(x)$. (h)–(j) Error of the reconstructed $y$-component of the force field, $F_y(x)$, obtained using BlinDNO, NIO, and FNO–NIO.
  • ...and 1 more figures

Theorems & Definitions (7)

  • Lemma 2.1
  • Definition 3.1: Function‐valued tuple
  • Definition 3.2: Permutation‐invariant set operator
  • Definition 3.3: Janossy pooling murphyjanossy
  • Definition 3.4: $k$-ary Janossy pooling murphyjanossy
  • Proposition 3.5
  • Proof 1