---
title: Potential Reconstruction in Inverse Problems
url: https://www.emergentmind.com/topics/potential-reconstruction
type: topic
---

# Potential Reconstruction in Inverse Problems

Searching arXiv for recent papers on “potential reconstruction” across domains to ground the article in published work.
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Searching a second time with a broader query.
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Potential reconstruction is the inference of an unknown scalar, vector, or effective potential from indirect observables rather than direct pointwise measurement. In the cited literature, it appears as an inverse boundary-value problem for the Schrödinger and wave equations, a model-independent recovery of dark-energy or inflationary potentials from cosmological observables, a reconstruction of cluster gravitational potentials from X-ray and thermal Sunyaev–Zel'dovich data, and an estimation of free-energy or electromagnetic vector potentials from stochastic trajectories and tomographic phase measurements [1204.0076][1702.04120][2112.09722][1802.07118][2002.12193][1704.06947]. The common structure is a forward map from a potential to data—boundary maps, spectra, phase shifts, rupture-time distributions, or expansion histories—and an inverse step that is typically ill-posed, model-dependent, or both.

## 1. Inverse-problem formulation

A recurring formulation starts from a PDE or SDE in which the potential enters as a coefficient. For stationary Schrödinger problems one encounters equations such as
$$
-\Delta \psi + v(x)\psi = E\psi
$$
in a bounded or unbounded domain, with data given by Dirichlet-to-Neumann or impedance boundary maps, Weyl functions, scattering amplitudes, or sampled eigenvalues [1204.0076][2307.13086]. For wave problems the unknown is a time-dependent coefficient in
$$
u'' - \Delta u + c\,u = f,
$$
and the data are full wave fields or sensor measurements derived from spatial-temporal averaging of the solution [1702.04120]. In stochastic settings the forward model is overdamped Langevin dynamics,
$$
dX = -\nabla U(X;\theta)\,dt + \sqrt{2D}\,dW,
$$
so that the object reconstructed from trajectories is an effective potential or free-energy landscape [2002.12193].

The same inverse logic persists in astrophysical and cosmological applications. In galaxy clusters, hydrostatic equilibrium,
$$
\nabla P = -\rho \nabla \Phi,
$$
together with a polytropic relation and spherical symmetry, turns deprojected X-ray emissivity or thermal SZ pressure into a reconstructed gravitational potential [2008.01107][1802.07118]. In late-time cosmology, Gaussian-process reconstructions of \(H(z)\), \(dH/dz\), or transverse comoving distance are converted into scalar-field potentials \(V(\phi)\) or dimensionless potentials \(U(\phi)\) through the Friedmann equations [2112.09722][2203.06767]. In inflation, the forward map is from \(V(\phi)\) to the primordial spectrum or to \(n_s(N)\), and the inverse step reconstructs \(V(\phi)\) from \(\mathcal P_{\mathcal R}(k)\) or from an ansatz for the scalar tilt [2109.14241][2505.10268].

This breadth suggests that “potential” is not a single ontological object but a role: it is the latent function that governs dynamics, transport, wave propagation, or equilibrium structure. The reconstruction problem is therefore defined as much by the data model and admissible function class as by the governing equations themselves.

## 2. Boundary, spectral, and scattering reconstructions

In inverse boundary problems, potential reconstruction is closely tied to uniqueness and identifiability. For a core-shell Schrödinger model in the unit disk with piecewise-constant potential, one Dirichlet-to-Neumann measurement uniquely determines the core potential coefficient when the radius is fixed, but uniqueness fails when both the potential and the core radius are allowed to vary; the nonuniqueness condition is expressed explicitly through a Bessel-function-based relation \(D(r_1,\alpha_1,r_2,\alpha_2)=0\) [1903.11825]. In a one-dimensional wave equation, if the potential \(q(x)\) is known on \((0,\varepsilon)\) near the accessible endpoint and the left boundary parameter \(h\) is known, a single measurement \(u(0,t)\) or \(u_x(0,t)\) determines the full potential \(q\), the length \(l\), and the terminal boundary parameter \(H\) via extraction of spectral data and Gelfand–Levitan theory [1901.02138].

A more general route is to convert boundary data into scattering data. For the Schrödinger equation at fixed energy, the impedance boundary map \(M_{\alpha,v}(E)\) can be related to generalized scattering amplitudes through an Alessandrini-type identity and a Fredholm boundary integral equation,
$$
[\psi(x,k)]_\alpha = [\psi^0(x,k)]_\alpha + \int_{\partial D} A_\alpha(x,y,k)\,[\psi(y,k)]_\alpha\,dy,
$$
after which known inverse-scattering machinery reconstructs the potential \(v\) [1204.0076]. This construction includes the Dirichlet-to-Neumann and Neumann-to-Dirichlet maps as special cases \(\alpha=0\) and \(\alpha=\pi/2\).

For complex-valued Sturm–Liouville potentials, "Reconstruction techniques for complex potentials" [2307.13086] replaces nonlinear inverse steps by two linear algebraic systems built from Neumann series of Bessel functions. The key recovery variable is
$$
Q(x):=\frac{q(x)}{4}-\frac{\omega^2(x)}{2},
$$
with final reconstruction
$$
q(x)=4Q(x)+2\omega^2(x).
$$
This avoids numerical differentiation of reconstructed coefficients and applies to inverse Weyl-function problems, inverse two-spectra problems, and inverse scattering on a finite interval.

Rough-potential reconstruction in two dimensions proceeds differently. Extending Bukhgeim’s quadratic-phase method, the boundary information
$$
BI_\lambda(x)= \frac{\lambda}{\pi} \Big\langle(\Lambda_q-\Lambda_0)[u_{\lambda,x}|_{\partial\Omega}], e^{i\lambda\overline{\psi}_x}|_{\partial\Omega}\Big\rangle
$$
is mollified with \(\sigma=\lambda^{-1/4}\), and for compactly supported \(q\in H^s(\mathbb R^2)\), \(s>0\),
$$
\lim_{\lambda\to\infty}\varphi_\sigma * BI_\lambda(x)=q(x)
$$
for almost every \(x\) [1811.09481]. This establishes reconstruction beyond classical smooth settings.

A more recent variant perturbs the medium itself. "The Calderon Problem Revisited: Reconstruction With Resonant Perturbations" [2307.12055] injects resonant droplets so that the effective coefficient becomes negative-valued with controllable amplitude, linearizes the resulting Neumann-to-Dirichlet map around \(\Delta-P^2\), and then reconstructs the original potential by CGO-based Fourier inversion. The method is asymptotically exact, with an explicit \( \mathcal O(P^{-\varsigma}) \) reconstruction error.

## 3. Cosmological and gravitational potential reconstruction

For galaxy clusters, the central claim is that gravitational potential can be reconstructed more robustly than hydrostatic mass. Using the non-radiative Omega500 simulation of 85 clusters, reconstructed potentials from X-ray and tSZ observables were compared with true simulated quantities. At \(R_{500}\), the hydrostatic mass has bias \(0.13\) and scatter \(0.24\), whereas the tSZ potential has bias \(0.06\) and scatter \(0.15\); the paper summarizes this as a potential bias of \(6\%\) versus a hydrostatic-mass bias of \(13\%\), with the scatter reduced by about \(35\%\) [2008.01107]. The reconstruction formulas are
$$
\Phi \propto [j_x]^{\eta}, \qquad \eta = \frac{2\Gamma - 1}{3+\Gamma},
$$
for X-ray emissivity, and
$$
\Phi \propto [P_{tSZ}]^{\eta}, \qquad \eta = \frac{\Gamma - 1}{\Gamma},
$$
for tSZ pressure. Substructures matter strongly: for the full sample at \(R_{500}\), the X-ray potential changes from \((0.31,0.38)\) with substructures to \((0.15,0.17)\) without them, while the tSZ potential changes from \((0.11,0.19)\) to \((0.06,0.15)\) [2008.01107].

The two-dimensional projected-cluster program complements this three-dimensional analysis. A Richardson–Lucy deprojection on a 2D grid is used to reconstruct projected gravitational potentials from X-ray surface brightness and SZ Compton-\(y\) maps under hydrostatic equilibrium and a polytropic gas law [1802.07118]. For X-rays, with bremsstrahlung-dominated emissivity \(j_x \propto T^{1/2}\rho^2\), the potential scales as
$$
\Phi \propto j_x^{1/\eta}, \qquad \eta=\frac{3+\gamma}{2(\gamma-1)},
$$
whereas for SZ pressure,
$$
\Phi \propto P^{1/\eta}, \qquad \eta=\frac{\gamma}{\gamma-1}.
$$
Applied to the simulated g1-cluster and to Abell 2142, the reconstructed X-ray and SZ projected potentials were found to be mutually consistent, and for Abell 2142 both reproduced well the projected potential inferred from lensing in the radial range covered by the data [1802.07118].

Dark-energy potential reconstruction uses cosmological expansion histories rather than cluster thermodynamics. One Gaussian-process approach reconstructs the dimensionless potential
$$
U(z)=E^2-\frac{E(1+z)}{3}\frac{dE}{dz} -\frac{\Omega_m(1+z)^3}{2} -\frac{2\Omega_k}{3}(1+z)^2,
$$
with \(E(z)=H(z)/H_0\), directly from \(H(z)\) data, and an analogous expression from supernova distance data \(D_M(z)\) [2112.09722]. The reconstruction depends on external priors; with the Planck 2018 \(3\sigma\) prior \(\Omega_k=-0.044\pm 0.050\), \(\Omega_m=0.315\pm 0.022\), the Peebles–Ratra power-law potential is within \(1\sigma\) for nearly \(60\%\) of the analyzed \(H(z)\) interval, while the free-field quadratic potential lies between the \(1\sigma\) and \(2\sigma\) bands for \(z\gtrsim 0.72\) [2112.09722]. A related GP reconstruction from 40 \(H(z)\) measurements writes
$$
V(z) = 3H^2 - (1+z)HH' - \frac{3}{2}H_0^2\Omega_0(1+z)^3,
$$
and
$$
\left(\phi'(z)\right)^2 = \frac{2H'H - 3H_0^2\Omega_0(1+z)^2}{(1+z)H^2},
$$
then eliminates \(z\) to obtain \(V(\phi)\). That analysis compares reconstructions obtained with a GP-estimated \(H_0\), a fixed Planck \(H_0=67.40\pm0.5\), and a fixed Hubble-mission \(H_0=73.52\pm1.62\), concluding that the inferred quintessence potential shifts substantially with the assumed \(H_0\) [2203.06767].

## 4. Spectrum-driven reconstruction in inflation and black-hole perturbation theory

In inflation, one direct strategy reconstructs the potential from the primordial scalar power spectrum under generalized slow roll. The reconstruction chain is
$$
\mathcal P_{\mathcal R}(k)\to f(\xi)\to H(\xi)\to \phi(\xi)\to V(\phi),
$$
with
$$
\log\!\left(\frac{1}{f^2}\right) = \int_0^\infty \frac{dk}{k}\,m(k\xi)\,\log\mathcal P_{\mathcal R}(k),
$$
$$
\frac{1}{H^2} = \frac{1}{H_i^2} -\frac{1}{(2\pi)^2}\int_{\xi_i}^{\xi} f^2(\log\xi')\,d\log\xi',
$$
and
$$
V = 3H^2\left[1-\frac{f^2H^2}{6(2\pi)^2}\right].
$$
The method is reliable when the GSR source function \(g\) remains small; for sharp or very large spectral peaks, such as \(A_p\sim 10^3\) or \(10^6\), the reconstructed spectrum deviates from the input although the location and qualitative height of the peak remain captured [2109.14241].

A complementary slow-roll reconstruction begins from an ACT-motivated tilt ansatz,
$$
n_s = 1 - \frac{p}{N+\alpha},
$$
derives
$$
\epsilon_V(N)=\frac{p-1}{2(N+\alpha)+C(N+\alpha)^p},
$$
and reconstructs an inverse-power plateau,
$$
V(\phi)=V_0\left[1-\left(\frac{M}{\phi}\right)^n\right], \qquad n=\frac{2(p-1)}{2-p},
$$
with tensor-to-scalar ratio
$$
r \approx \frac{16(p-1)}{C (N+\alpha)^p}.
$$
The integer cases \(n=1,2,3,4,5\) correspond to \(p=4/3,3/2,8/5,5/3,12/7\) and are highlighted as ACT-compatible benchmarks [2505.10268].

In black-hole perturbation theory, potential reconstruction is formulated as a theory-agnostic inverse problem from quasi-normal modes. Small deviations from GR are expanded as
$$
\delta V_{ij} = \frac{1}{H^2}\sum_{k=0}^{\infty}\alpha^{(k)}_{ij}\left(\frac{H}{r}\right)^k,
$$
and the QNM frequencies are expanded to second order in these coefficients. Bayesian fitting plus principal component analysis then identifies the combinations of parameters actually constrained by the data [2202.08655]. Percent-level measurements of the fundamental mode and first two overtones can constrain the effective tensor potential and some coupling combinations, but the inverse problem remains degenerate: potential-versus-coupling ambiguities, exchange symmetry between \(V_{01}\) and \(V_{10}\), and loss of sensitivity far from the light ring limit the reconstruction. The paper’s summary is that two to three robust PCA features, rather than a full function, are typically recoverable [2202.08655].

## 5. Effective, free-energy, and vector potentials from trajectories and tomography

When the data are trajectories rather than boundary traces, the inferred object is often an effective potential. In "Reconstruction of effective potential from statistical analysis of dynamic trajectories" [2002.12193], observed atomic displacements are modeled by overdamped Langevin dynamics and a Bayesian posterior over potential parameters is approximated by sequential Monte Carlo. The graphene silicon-vacancy example assumes
$$
U(x,y,z;\theta)=U(x,y)+U(z),
$$
with
$$
U(x,y)=a(x+y)\left[1+b\cos\!\left(4\tan^{-1}(x/y)\right)\right], \qquad U(z)=cz+dz^2.
$$
Using 1141 observations separated by \(\delta=0.025\) ps, the posterior mean effective potential is reported as
$$
V(x,y)=2.06(x+y)\left[1+0.0125\cos\!\left(4\tan^{-1}(x/y)\right)\right], \qquad V(z)=0.039\,z+0.021\,z^2
$$
[2002.12193].

Potential reconstruction from strong-field photoelectron spectra uses a different inverse loop. A one-dimensional time-dependent Schrödinger equation in velocity gauge is solved repeatedly while an optimization algorithm varies the single-active-electron potential until the photoelectron momentum distribution matches a target [2012.05179]. The paper considers both grid-based and parametric representations, with objective
$$
\rho_{2}\left[f,g\right]=\left\{\int_{a}^{b}dx\left[f\left(x\right)-g\left(x\right)\right]^{2}\right\}^{1/2}.
$$
For the soft-core Coulomb test problem, reported minima include \(\rho_2\approx 0.019\), \(\rho_2\approx 0.03\), and, with a denser grid, \(\rho_2=0.0042\). If the target PMD is generated at \(2.0\times10^{14}\,\mathrm{W/cm^2}\) but optimization assumes \(1.0\times10^{14}\,\mathrm{W/cm^2}\), reconstruction degrades to \(\rho_2\approx 0.61\); allowing the field strength to vary recovers \(F_0=0.0763\) a.u. versus exact \(0.0755\) a.u., with \(\rho_2\approx 0.032\) [2012.05179].

Electromagnetic-potential reconstruction in microscopy is formulated as a tomographic MAP problem. In Lorentz TEM, the phase shift contains an Aharonov–Bohm contribution from the magnetic vector potential,
$$
\varphi_{m}(\mathbf{r}_{\perp})=-\frac{e}{\hslash}\int_L \mathbf{A}(\mathbf{r}_{\perp},z)\cdot d\mathbf{r},
$$
and two orthogonal tilt series generate Fourier-slice relations for \(\tilde A_x,\tilde A_y,\tilde A_z\) [1704.06947]. "3D Reconstruction of the Magnetic Vector Potential using Model Based Iterative Reconstruction" [1704.06947] replaces filtered back projection by a MAP estimator with a \(q\)-GGMRF prior and iterative coordinate descent. On simulated nanoparticles, missing-wedge VFET reconstructions are about \(5\%\)–\(25\%\) worse in NRMSE than full-range data, while MBIR reduces planar NRMSE for \(A_x\) and \(A_z\) by about \(10\%\)–\(20\%\) in many planes, and by about \(10\%\) across all three components for the cylindrical nanoparticle. On experimental permalloy data, the MBIR reconstruction converges in 35 iterations on a \(256^3\) voxel grid [1704.06947].

A more formal quantum-mechanical variant reconstructs a real potential directly from a complex eigenfunction. If \(\psi(x)=R(x)e^{iS(x)}\) is nodeless and
$$
2R'S'+RS''=0, \qquad S(x)=C\int^x \frac{ds}{R^2(s)},
$$
then
$$
V(x)=E+\frac{R''}{2R}-\frac{C^2}{2R^4}.
$$
The reconstructed state carries constant current \(j_x=R^2S'=C\), so the method yields localized scattering states rather than ordinary bound states in the examples discussed [1704.03825].

## 6. Ill-posedness, regularization, and recurring limitations

Across domains, potential reconstruction is typically ill-conditioned. In rupture-time inversion, a single first-passage-time distribution reliably reconstructs only coarse features such as barrier height, barrier width, and very low-dimensional shapes; the paper reports that three parameters are typically reconstructible reliably, five are marginal, and six are usually too many for a single FPTD [1002.2990]. The inverse condition number is best as the effective Peclet number \(A^*\to 0\), and for constant weighting the best conditioning typically occurs near an initial position \(x\approx 0.75\). More complex multi-minima potentials become reconstructible only after combining two or more measured FPT distributions obtained under different temperatures, initial positions, or known probe forces [1002.2990].

For time-dependent wave potentials, the derivative of the parameter-to-solution map is compact with infinite-dimensional range, so the linearized inverse problem is locally ill-posed [1702.04120]. The remedy there is successive linearization with REGINN and discrepancy-principle stopping. In boundary-value and quantum problems, Tikhonov regularization, Newton iterations, and Fredholm integral equations serve the same role [1903.11825][1204.0076]. In trajectory-based and tomographic settings, Bayesian sequential Monte Carlo and MAP estimation with explicit priors are the dominant stabilizers [2002.12193][1704.06947]. In cosmology, the regularization is often implicit in Gaussian-process kernels and external priors on \(\Omega_m\), \(\Omega_k\), or \(H_0\) [2112.09722][2203.06767].

A second recurring issue is structural degeneracy. Fixed geometry can make one-measurement recovery unique, while allowing shape variation can destroy uniqueness [1903.11825]. In cluster reconstruction, substructures, non-thermal pressure support, triaxiality, and the choice of substructure-removal procedure materially change biases and scatter [2008.01107]. In black-hole spectroscopy, QNMs alone generally constrain only a few principal combinations of potential and coupling parameters rather than the underlying theory [2202.08655]. In dark-energy reconstruction, compatibility of candidate potentials depends strongly on prior choices and redshift range [2112.09722].

These results suggest a consistent methodological lesson: potential reconstruction improves when the forward model is physically explicit, the admissible potential class is constrained, and complementary datasets or controlled perturbations are introduced. The literature repeatedly replaces a brittle one-shot inversion by a staged procedure—boundary map to scattering data, spectrum to background function, deprojection to thermodynamic variable, or trajectory to diffusion posterior—because those intermediate representations are where uniqueness, regularization, and uncertainty quantification become tractable.

Source: https://www.emergentmind.com/topics/potential-reconstruction