Initial-to-Final-State Inverse Problem

• The initial-to-final-state inverse problem is defined as recovering the unknown initial configuration of a dynamical system from its later or final state measurements.
• It spans diverse fields such as quantum mechanics, cosmology, and fluid dynamics, and is challenged by nonuniqueness and instability due to nonlinearity and dissipation.
• Advanced techniques like Tikhonov regularization, Carleman estimates, and neural network inversion offer practical methods to mitigate ill-posedness and improve reconstruction fidelity.

The initial-to-final-state inverse problem concerns the recovery of an unknown initial configuration of a dynamical system from measurements at a final or later state, with the dynamics governed by a (typically nonlinear and dissipative or chaotic) forward evolution. This problem appears across mathematical physics, cosmology, fluid dynamics, quantum mechanics, and PDE theory, and involves fundamental challenges related to non-uniqueness, instability, and the development of specialized reconstruction algorithms. The following article details the mathematical principles, exemplary methodologies, and implications of this inverse problem class—focusing on representative research in nonlinear macro- and microscale dynamics, cosmological field theory, reaction-diffusion, and quantum inverse problems.

1. Problem Formulation and Mathematical Setting

The initial-to-final-state inverse problem (sometimes called a final value, backward, or terminal data problem) asks: given an evolution law $u'(t) = F(u(t))$ (ODE, PDE, or map; $u$ in a suitable Hilbert or Banach space), find the initial state $u_0$ so that at some fixed later time $T$ one has $u(T;u_0) = g$, where $g$ is the observed (“final”) state.

Abstract Setting

• Forward map: $\Phi: u_0 \mapsto u(T;u_0)$.
• Inverse problem: Given $g$, find $u_0$ with $\Phi(u_0) = g$.
• Nonlinearity and ill-posedness: In dissipative systems, $\Phi$ severely contracts information; for strongly nonlinear or chaotic flows, $\Phi$ may not be injective, and small errors in $g$ can lead to large (or unbounded) variations in the reconstructed $u_0$. This ill-posedness is fundamentally characterized by instability and—for many models—nonuniqueness.

Paradigmatic Equations

• ODEs: $y'(t)=f(t,y(t)),\quad y(T)=y_T$ (final value problem, see (Wang et al., 2018)).
• Parabolic PDEs: $u_t=Au + f(u)$, reconstruct $u_0$ from $u(T)$ (BenSalah et al., 6 Dec 2024, Ngoc et al., 2019, Imanuvilov et al., 2022, Hassi et al., 2021).
• Fluid dynamics: inverse Navier–Stokes and Brinkman–Forchheimer, identify initial velocity or source from data at $t=T$ (Kumar et al., 2021, Kumar et al., 2023).
• Quantum evolution: Schrödinger with general potential, reconstruct $V$ from the operator mapping $u_0 \mapsto u(T)$ (Cañizares et al., 8 Mar 2025, Caro et al., 4 Dec 2025, Caro et al., 2023, Chen et al., 2020).
• Cosmological structure formation: recover early Universe linear displacement field from nonlinear evolved state (Jindal et al., 2023).
• Hamilton–Jacobi: construct all $u_0$ so $u(T;u_0)=u_T$ (Esteve et al., 2020).

2. Uniqueness, Nonuniqueness, and Structural Barriers

The uniqueness of recovering $u_0$ (or a generating parameter, such as a quantum Hamiltonian or initial density field) from the final state fundamentally depends on properties of the dynamics and the nature of the observable $g$.

• Finite-dimensional ODEs: For Lipschitz $f$, final value problems are well-posed for short time or strictly monotone flows, yielding unique backward solutions (Wang et al., 2018). For strongly contracting systems or in the presence of bifurcations, uniqueness fails.
• Parabolic PDEs: For dissipative semigroups, $\Phi$ is compact and non-injective; recovery of $u_0$ is ill-posed and possible only with regularization. Uniqueness holds within restricted, smooth subspaces (e.g., for analytic data or under final state observability), but reconstructions are logarithmically unstable (Hassi et al., 2021, BenSalah et al., 6 Dec 2024).
• Nonlinear/chaotic flows: In systems like cosmological N-body evolution, the mapping from initial to final state is “one-to-many” on small scales—many $x_{\text{init}}$ yield nearly indistinguishable $x_{\text{final}}$ due to orbit mixing in halos or other chaotic regions (Jindal et al., 2023).
• Quantum mechanics: For Schrödinger evolution, knowledge of the operator $\mathcal{U}_T$ for all initial data determines the potential $V(x)$ uniquely (up to gauge), even with time-dependence and under minimal decay (Caro et al., 2023, Caro et al., 4 Dec 2025, Cañizares et al., 8 Mar 2025). For classical Hamiltonian systems, uniqueness holds generically via Liouville inversion (Chen et al., 2020).
• Nonlinear Hamilton–Jacobi: The inverse map is nonunique; when reachable, all initial data coincide with the minimal preimage on the “active set,” and elsewhere constitute an infinite-dimensional family above it (Esteve et al., 2020).

3. Analytical Frameworks and Stability Mechanisms

Several analytical constructions anchor both theoretical and practical treatments. Stability, regularization, and inversion strategies depend on dynamical type and data structures.

Regularization and Conditional Stability

• Tikhonov or Morozov Regularization: For ill-posed parabolic/fractional PDEs, reconstruction is framed as a minimization problem:

$$\min_{g} \|A(g) - h^\delta\|^2_{L^2} + \alpha \|g\|^2_{L^2},$$

where $A(g)$ is the solution map to final time and $\alpha > 0$ penalizes large initial conditions (BenSalah et al., 6 Dec 2024, Coşkun, 2020).

• Carleman and logarithmic convexity estimates: Analytical stability is often quantified via inequalities of logarithmic type: for analytic semigroups, initial norm $\leq C |\ln(\|\text{final data}\|)|^{-\beta}$ (Hassi et al., 2021, Imanuvilov et al., 2022). These logarithmic rates are optimal for parabolic evolution.
• Fixed-point and Schauder/Tikhonov approaches: For nonlinear flows, existence and continuous dependence are constructed via mapping the inverse problem to a contraction or compact operator equation in suitable function spaces (Kumar et al., 2021, Ngoc et al., 2019, Kumar et al., 2023).
• Interpolation and interpolation inequalities: For stochastic or controlled systems, advanced Carleman-based interpolation inequalities bridge initial and final data, supporting conditional (often weak, e.g., power-type) stability (Elgrou et al., 13 Oct 2024).

Posterior Approaches for Chaotic Dynamics

In “one-to-many” inverse mappings, as in cosmological structure formation, deterministic methods trained on forward–inverse pairs (e.g., V-Net neural inversion) select a physically plausible “mode-like” solution, but reconstruct only the dominant branch determined by the training data mean (Jindal et al., 2023).

4. Computational and Algorithmic Approaches

Given the analytical ill-posedness, practical inversion depends on regularization, robust optimization, and efficient surrogate modeling, especially in high-dimensional fields and when the forward operator is expensive.

• Gradient-based variational methods: Minimization of Tikhonov or similar functionals, often via conjugate-gradient descent (with gradients computed via an adjoint PDE), is standard for parabolic, diffusion, and heat-like problems (Coşkun, 2020, BenSalah et al., 6 Dec 2024).
• Neural network inversion: In cosmology, deterministic convolutional architectures (e.g., V-Net) trained by supervised regression can approximate $f^{-1}$ accurately up to the non-chaotic regime, bypassing expensive Monte Carlo and adjoint optimization (Jindal et al., 2023). Loss is typically MSE between predicted and true initial fields.
• Stochastic/optimal control and Nash equilibria: For stochastic reaction-diffusion with random boundary and control, coupled forward-backward SPDEs and Nash equilibrium conditions yield reconstructors of initial data under multi-objective optimality criteria (Elgrou et al., 13 Oct 2024).
• Spectral/Abel integral inversion: For certain nonlinear wave equations (e.g., 1D tsunami SWE), explicit hodograph and Abel integral transforms permit exact separation of initial velocity and displacement from final shoreline dynamics, exploiting linearization of the governing system (Rybkin et al., 27 Jan 2025).
• Carleman-based reconstructors: In coefficient inverse problems and MFG systems, Carleman inequalities and boundary overdetermination controls permit the explicit recovery (and stability estimation) of spatial coefficients from final and lateral data (Klibanov et al., 2023, Imanuvilov et al., 2022).

5. Instabilities, Limitations, and Nonuniqueness Phenomena

• Chaotic irreversibility: On small scales and late times, deterministic inversion cannot recover fine-grained features (e.g., individual halo seedings in cosmology); only large-scale modes or ensemble averages are stably reconstructible (Jindal et al., 2023).
• Multiplicity of admissible pre-images: For nonlinear hyperbolic flows (Hamilton–Jacobi), the set of pre-images is typically infinite-dimensional and characterized by local minimality on a well-defined “active set,” with arbitrary excursions elsewhere (Esteve et al., 2020).
• Nonuniqueness in partial data: For parabolic and hyperbolic PDEs, uniqueness may require spatially extended or full data (e.g., entire boundary and final/initial observations), or fail without sufficient geometric information (Imanuvilov et al., 2022).
• No uniform (e.g., Hölder) stability in generic dissipative flows: Despite uniqueness, logarithmic stability bounds are optimal due to the compactness of the parabolic semigroup (Hassi et al., 2021); for fractional and degenerate equations, ill-posedness can be more severe (BenSalah et al., 6 Dec 2024).

6. Key Research Examples and Methodological Innovations

The following table collects several representative settings, inversion types, and principal techniques as found in the modern arXiv literature:

System/PDE Type	Reconstruction Objective	Principal Theoretical/Algorithmic Tool
Cosmological N-body (V-Net)	Initial linear field from nonlinear state	3D CNN regression, percent-level MSE loss (Jindal et al., 2023)
Fractional reaction-diffusion	Initial data from final state	Mittag-Leffler operators, Banach fixed-point (Ngoc et al., 2019)
Heat/conduction, source+state	Simultaneous source and $u_0$ reconstruction	Tikhonov+conjugate gradient, adjoint PDE (Coşkun, 2020)
Ornstein-Uhlenbeck (parabolic)	Initial data from partial final observables	Logarithmic convexity, analytic semigroup (Hassi et al., 2021)
Quantum (Schrödinger)	Potential $V$ from $\mathcal{U}_T$	CGO, stationary state, Strichartz, Fourier arguments (Caro et al., 2023, Caro et al., 4 Dec 2025, Cañizares et al., 8 Mar 2025)
1D nonlinear SWE (tsunami)	$\eta_0(x),u_0(x)$ from shoreline motion	Carrier-Greenspan transform, Abel inversion (Rybkin et al., 27 Jan 2025)
MFG/mean field games	Spatial kernel $b(x)$ from overdetermined data	Carleman estimates, final data integration (Klibanov et al., 2023)
Stochastic reaction-diffusion	$(y_0,y_{0,\Gamma})$ from $(y(T),y_\Gamma(T))$	Interpolatory Carleman inequalities, Nash control (Elgrou et al., 13 Oct 2024)

7. Broader Implications and Future Directions

Understanding and effectively addressing the initial-to-final-state inverse problem underpins numerous applied and theoretical domains:

• Physical cosmology: Enables statistical inference of primordial structure, providing necessary initial conditions for constrained forward simulations and uncertainty quantification pipelines (Jindal et al., 2023).
• Quantum information/data-driven dynamics: Validates the sufficiency of input–output mappings (the initial-to-final propagator) for full system identification, and motivates principled algorithms for Hamiltonian learning (Caro et al., 2023, Caro et al., 4 Dec 2025).
• Reaction-diffusion and control: Informs optimal sensor placement, feedback stabilization, and robust control for parabolic and fractional systems (BenSalah et al., 6 Dec 2024, Coşkun, 2020, Elgrou et al., 13 Oct 2024).
• Mathematical theory: Sharpens the boundary between identifiability and severe nonuniqueness, quantifies stability in terms of dynamical regularity (analyticity, hyperbolicity), and underlines the centrality of Carleman and functional-analytic estimates (Hassi et al., 2021, Imanuvilov et al., 2022, Ngoc et al., 2019).
• Algorithmic acceleration: Neural surrogates and regression inversion, especially in high-dimensional fields, provide tractable and rapid approximate inverses where traditional sampling or adjoint techniques are infeasible (Jindal et al., 2023).

Open directions include robust uncertainty quantification for multimodal and stochastic inverse maps (e.g., Bayesian and generative diffusion models), extension of uniqueness and stability to less restrictive data classes and more general geometric configurations, and adaptation of the present frameworks to coupled multiphysics and high-noise regimes.

Theoretical advances in analytic and geometric control, data-driven regularization, and hierarchical statistical inversion will continue to deepen both the understanding and computational tractability of the initial-to-final-state inverse paradigm across the sciences.