Determining evolutionary equations from a single passive boundary observation

TL;DR

This paper develops a unified mathematical framework to recover unknown sources and coefficients in evolutionary PDEs using only a single passive boundary observation, addressing a largely open inverse problem.

Contribution

It introduces a systematic method combining integral identities and analysis to solve inverse boundary problems with minimal data for various PDEs, surpassing prior approaches.

Findings

First systematic resolution for hyperbolic, parabolic, and Schrödinger equations using passive data.

Measurement data must exceed unknowns by at least one dimension for unique recovery.

Framework subsumes existing results and applies to more general practical configurations.

Abstract

We study inverse boundary problems for evolutionary PDEs using only a single passive boundary observation, where data from an unknown internal source propagate through an unknown medium without active inputs. The goal is the simultaneous recovery of coupled unknowns (sources and coefficients) from severely limited data. Unlike active methods with rich, structured inputs, passive observation poses two core challenges: minimal information and intrinsic coupling of multiple unknowns. Consequently, such problems remain largely open and unsystematically studied. We develop a unified framework based on integral identities, harmonic and microlocal analysis, and low-/high-frequency asymptotics. This approach yields the first systematic resolution for second-order hyperbolic, parabolic, and Schr\"odinger equations under a single coherent method. The key condition requires the measurement dataset's cardinality to exceed the unknowns' by at least one dimension, providing room to decouple unknowns and linearize the nonlinear inverse problem. Our unique identifiability results subsume all existing literature and cover more general configurations of practical interest. This framework complements classical theories and opens a promising new direction for future development.