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Structure-Driven Inversion: A New Paradigm for Solving Inverse Problems

Shengchang Chen

arXiv:2609.16536v2math.NAcs.NAphysics.geo-ph

TL;DR

Inverse problems are often ill-posed, and optimization-driven methods may overlook their intrinsic structure. This paper proposes constructing inversion methods from mathematical or physical structure, with SDI framed as a complement to ODI for problems with identifiable structure. Its conclusion presents SDI as a customized direct strategy, not a replacement for ODI.

  • Problem

    Inverse problems are often ill-posed, and optimization-driven methods can overlook special structure that may support well-posed inversion.

    Source: §1 · p. 1, §1.2 · p. 3

  • Method

    SDI identifies and exploits intrinsic mathematical or physical structure to construct a solution method, with representative approaches spanning pseudo-inversion, virtual-source projection, and structural projections.

    Source: §1.2 · p. 4, §11 · p. 16

  • Results

    The paper concludes that SDI complements ODI: it offers a structure-customized path for identifiable structure, while ODI remains universal for problems with missing or incomplete structure.

    Source: §10 · p. 15, §11 · p. 16

  • Takeaways & Limitations

    SDI frames inversion as moving from objective-function optimization toward identifying problem structure and constructing a structure-customized solution method.

    Source: §11 · p. 16

    The proposed SDI path is scoped to inverse problems with identifiable special structure; the paper describes inverse problems as mostly, not universally, ill-posed.

    Source: §1.2 · p. 3

Abstract

from arXiv · show

Inverse problems are predominantly solved within the optimization-driven paradigm, which formulates the problem as objective-function minimization and approaches the solution by iterative search. Though universal, it suffers from limitations in efficiency, interpretability, and multi-parameter decoupling. This paper proposes a new paradigm---Structure-Driven Inversion (SDI). Instead of iterative search, SDI identifies and exploits the intrinsic structure of the problem to construct a solution method. It has two types of structure---mathematical structure and physical structure---and two corresponding driving inversion types: Mathematical Structure-Driven Inversion (MSDI) and Physical Structure-Driven Inversion (PSDI). The current representative method of MSDI is Mathematical Structure-Driven Pseudo-Inverse Inversion (MSDPII), which constructs a pseudo-inverse in the spectral domain via unitary diagonalization. The current representative methods of PSDI are Physical Structure-Driven Waveform Inversion Imaging (PSDWII), which performs inversion through virtual-source projection, and Physical Structure-Driven Back-Propagation (PSDBP), which implements three structural projections for deep neural networks via physical-system analogy. SDI is not a rejection but a complement and extension of Optimization-Driven Inversion (ODI). To the best of the author's knowledge, no existing study has systematically presented structure-driven inversion as an independent paradigm; this paper aims to fill that gap.

1 Introduction

The paper introduces structure-driven inversion as a complement to optimization-driven methods, which are widely used across inverse-problem domains but may overlook problem-specific structure. It proposes constructing solution methods from identifiable mathematical or physical structure.

1.1 Universality of the Optimization-Driven Paradigm: Seven Representative Classes of Inverse Problems

  • Across seven representative classes, inverse-problem methods share the pattern of constructing an objective function and solving it iteratively despite differing physical backgrounds and forward operators.The classes span geophysical inversion, image reconstruction, parameter identification, inverse scattering, optimal control, system identification, and machine-learning parameter estimation.Source: §1.1 · p. 2, Table 1 · p. 3

1.2 Achievements and Limitations of the Optimization-Driven Paradigm

  • Treating structured operators and physical processes as general mappings can compromise efficiency, interpretability, and multi-parameter decoupling within the optimization-driven paradigm.The paper also notes that the adjoint wavefield's physical meaning remains unclear in the adjoint-state method.Source: §1.2 · p. 3
  • Some inverse problems with special structure, such as unitary or completely diagonalizable forward operators, can be well-posed and motivate structural construction instead of blind search.The paper describes inverse problems as mostly, but not universally, ill-posed.Source: §1.2 · pp. 3–4

1.3 Positioning of This Paper: A New Paradigm

  • SDI is presented as complementary to ODI: it targets problems with identifiable structure, while ODI retains universal solving capability when structure is missing or incomplete.The paper reports that, to the author's knowledge, no prior study systematically presented SDI as an independent methodological paradigm.Source: §1.3 · pp. 4–5
  • SDI constructs solution methods from mathematical or physical structure, with MSDI and PSDI represented by MSDPII, PSDWII, and PSDBP.These methods respectively use spectral-domain pseudo-inversion, virtual-source projection, and structural projections through physical-system analogy.Source: §1.3 · p. 4

2 Related Work

The related-work review distinguishes methods that use structural information within optimization from SDI, which uses structure to construct the solution method itself.

2.1 Regularization Methods

  • Tikhonov and total-variation regularization use mathematical structure as a prior or penalty within optimization, without changing its objective-function and iterative-solving logic.Tikhonov regularization stabilizes solutions through a penalty term, while TV regularization exploits image-gradient sparsity.Source: §2.1 · p. 5

2.2 Statistical and Bayesian Inversion

  • Bayesian inversion represents prior knowledge with a probability distribution, while posterior maximization or sampling still relies on iterative optimization or MCMC.Source: §2.2 · p. 5

2.3 Physics-Informed Neural Networks (PINNs)

  • PINNs place physical-equation residuals in the loss function and train through back-propagation and gradient descent, so physical information acts as a constraint rather than a basis for constructing the solution method.Source: §2.3 · p. 5

2.4 Structurally Constrained Inversion and Joint Inversion

  • Structurally constrained and joint inversion use geological or cross-field structural information as constraints within the optimization framework.Source: §2.4 · p. 5

2.5 Model Reduction and Operator Learning

  • Model reduction and operator learning exploit low-dimensional or learned mappings to accelerate computation or replace forward modeling, rather than changing the inversion solution paradigm.SDI instead uses the mathematical or physical structure of the forward operator or system response to construct the solution method.Source: §2.5 · p. 5

3 The Optimization-Driven Paradigm

Optimization-driven inversion formulates inverse problems as objective-function minimization and solves them through iterative search. Its gradient computation can couple parameters, while the paradigm faces efficiency, interpretability, and decoupling limitations.

3.1 Unified Formulation of the Paradigm

  • From Tikhonov (1960s) to Adam (2010s), methods show technological progress within the same paradigm—richer algorithms, larger scales, and greater automation—while methodological logic remains unchanged.Source: §3.1 · p. 6

3.2 Core Operation of ODI: Cross-Correlation (Multiplication)

  • Cross-correlation multiplies sensitivity by residual to form the gradient direction, mixing contributions from different parameters.This mathematical coupling can require Hessian inversion for decoupling, at a computational cost of O(N^3).Source: §3.2 · p. 6

3.3 Three Inherent Limitations of ODI

  • ODI requires repeated full forward computations, yields gradients with ambiguous physical meaning, and couples parameter contributions during gradient computation.Tens to hundreds of iterations are typical; Hessian inversion to decouple parameters incurs enormous computational cost.Source: §3.3 · p. 6

4 The Structure-Driven Paradigm

Structure-Driven Inversion (SDI) identifies and exploits intrinsic problem structure to construct a solution method. It complements optimization-driven inversion, with mathematical and physical structure as distinct routes.

4.1 Core Definition

  • SDI uses identifiable invariance, decomposability, or hierarchy in a forward operator or system response to construct a solution method.The paper distinguishes mathematical structure from physical structure, including causal-chain decomposition.Source: §4.1 · p. 7

4.4 Philosophical Divide Between the Two Paradigms

  • ODI and SDI share the goal of fitting data but differ in their principal route: iterative search versus structural construction.The paper presents SDI as a complement and extension, not a rejection, of ODI.Source: §4.1 · p. 7, Table 5 · p. 8, §4.4 · p. 8

4.2 Two Driving Inversion Types and Open Method Development

  • SDI has two types, MSDI and PSDI, with MSDPII, PSDWII, and PSDBP as current representative methods; the framework remains open to new methods.MSDI exploits operator algebra, while PSDI exploits a physical process's causal chain, directly or by analogy for neural networks.Source: §4.2 · p. 7, §4.1 · p. 7

4.3 Core Operation: Division/Inversion/Deconvolution

  • Division, inversion, and deconvolution separate coupled contributions and directly extract target parameters without using the Hessian matrix.These operations are identified as the common effect of the representative methods' core operations.Source: §4.3 · p. 7, Table 4 · p. 8

5 Mathematical Structure-Driven Pseudo-Inverse Inversion (MSDPII): Spectral-Domain Pseudo-Inverse

MSDPII constructs a pseudo-inverse from the operator's spectral structure rather than relying on numerical SVD computation. Its applicability is limited to suitable square operators, including some approximately diagonalizable cases with controllable error.

5.1 Structure Identification

  • For G = TᴴΛT, translation-invariant convolution-type operators satisfy the unitary-diagonalization condition, with the discrete Fourier transform as a special case [35] [36].Here T is unitary and Λ is diagonal.Source: §5.1 · p. 8

5.2 Structure Exploitation: Analytical SVD

  • The analytical SVD derives U, Σ, and Vᴴ directly from the spectral decomposition, so no numerical SVD computation is required.The construction uses the phase diagonal matrix P and singular values |λk|.Source: §5.2 · p. 8

5.3 Structure-Driven Inverse Operator

  • The regularized spectral filter is bounded by 1/(2√α) and approaches 1/λk for nonzero λk as α→0+.This is the spectral-domain pseudo-inverse construction described for MSDPII.Source: §5.3 · p. 9

5.4 Comparison with ODI

  • MSDPII and Tikhonov regularization are numerically equivalent but methodologically different: MSDPII starts from matrix structure rather than an optimization problem.Source: §5.4 · p. 9, Table 6 · p. 9

5.5 Applicability and Limitations

  • MSDPII covers translation-invariant systems, periodic-boundary PDEs, symmetric integral equations, and potential-field inversion, but requires square, unitarily diagonalizable G.Approximately unitarily diagonalizable operators may also be used when approximation error is controllable.Source: §5.5 · p. 9

6 Physical Structure-Driven Waveform Inversion Imaging (PSDWII): Virtual-Source Inversion

PSDWII reverses the physical causal chain of wave propagation through virtual-source projection rather than iterative gradient descent. The section describes its source structure, inversion steps, nonlinear handling, and applicability boundaries.

6 Physical Structure-Driven Waveform Inversion Imaging (PSDWII): Virtual-Source Inversion

  • PSDWII approximates inversion through structural projection, unlike the adjoint-state method’s iterative gradient descent.Both involve reverse tracking of the physical process, but use different operational paths.Source: §6 · p. 9

6.1 Structure Identification: Causal Chain of Wave Propagation

  • Wave propagation is organized as two propagations and one virtual source, which physically links observed data with model parameters [38].The virtual source is the product of incident-wave interaction with heterogeneity, not a direct data-to-model mapping.Source: §6.1 · p. 10

6.2 Mathematical Formulation of the Virtual Source

  • The formulation introduces the background wavefield under model m0 and the virtual-source relation, with weak scattering treated under the Born approximation.The supplied passages do not state the equations’ specific expressions.Source: §6.2 · p. 10

6.3 Physical Classification of Virtual Sources

  • Virtual sources are classified by heterogeneity scale relative to wavelength, while all three share the incident-wave interaction and secondary-wave generation structure.The passages do not specify the individual classes’ properties.Source: §6.3 · p. 10, Table 7 · p. 10

6.4 Virtual-Source Inversion

  • Inversion first adjoint-projects observed data to extract the virtual source, then deconvolves incident-wave and radiation-pattern effects.A small parameter ϵ may be added to the denominator when it approaches zero to avoid numerical instability.Source: §6.4 · pp. 10–11

6.5 Handling of Nonlinear Problems: Iteration Under Structural Constraints

  • Strong multiple scattering requires stepwise inversion: update the background model, recompute forward modeling, extract a new virtual source, and repeat to the residual requirement [38].This structural iteration differs from optimization-driven iteration, as compared in Table 8.Source: §6.5 · p. 11, Table 8 · p. 11

6.6 Applicability and Limitations

  • PSDWII applies to wave-type problems with clear causal chains, including seismic FWI, inverse scattering, least-squares RTM, and RTM.These examples are listed as applications in the section.Source: §6.6 · p. 11
  • It requires a sufficiently accurate background model and Born-valid initial-model deviations; strongly nonlinear problems still require multiple iterations.Insufficient kinematic accuracy can cause cycle skipping.Source: §6.6 · p. 11

7 Physical Structure-Driven Back-Propagation (PSDBP): Three Structural Projections

PSDBP applies a physical-system analogy to layered neural networks and constructs three structural projections. Its scope includes layered network architectures, with requirements on activation invertibility and stability and a need for further empirical verification.

7 Physical Structure-Driven Back-Propagation (PSDBP): Three Structural Projections

  • PSDBP transfers the physical-structure-driven idea of PSDWII to deep learning by treating a layered neural network as an analogous physical system.Inter-layer connections correspond to stiffness matrices, activations to constitutive relations, and forward propagation to excitation-response.Source: §7 · p. 12

7.1 Structure Identification in Deep Neural Networks

  • The network’s identifiable layered structure provides the basis for the physical-system analogy.The section also presents a structural correspondence table.Source: §7.1 · p. 12, Table 9 · p. 12

7.2 Three Structural Projections

  • The three projections transport adjoint demand through connection geometry without activation derivatives, invert constitutive relations analytically, and update stiffness by input-residual correlation.The constitutive inversion uses σ−1 instead of σ′.Source: §7.2 · p. 12

7.4 Applicability and Limitations

  • PSDBP is described for layered networks including fully connected, convolutional, residual, and recurrent architectures.These are the listed applicable network types.Source: §7.4 · pp. 12–13
  • The activation must be strictly monotonic and invertible, its analytical inverse needs stability handling during depth propagation, and the theoretical analysis needs further empirical verification.Source: §7.4 · p. 13

8 Unified Framework of SDI

The unified framework describes inversion as reversing a structured forward process, with local inversion between undoing two substructures. SDI’s iterative form and applicability depend on available structure, and it complements rather than replaces ODI.

8.1 Unified Mathematical Formulation of the Two Driving Inversion Types

  • The shared framework maps model through two substructures and a local action to data; inversion reverses the substructures around a local inversion.The forward local action is multiplication, while local inversion is division or deconvolution.Source: §8.1 · p. 13

8.2 Iteration View of SDI

  • SDI may iterate depending on the nature and completeness of available structure, but its iteration is structural projection rather than blind search.Source: §8.2 · p. 13

8.3 Applicability Conditions and Boundaries

  • SDI applies when operators can be unitarily or approximately unitarily diagonalized, or when physical causal chains or structural analogies are available; otherwise it degenerates to ODI.The framework presents SDI and ODI as complementary, with ODI retaining universal applicability for unstructured problems.Source: §8.3 · pp. 13–14, Table 11 · p. 14

9 Comparison of SDI and ODI

The paper positions SDI as a paradigm shift that complements ODI rather than substituting for it. It frames the extension as moving from data-fitting algorithms toward co-design of structure and algorithm.

9.1 Systematic Comparison at the Paradigm Level

  • Table 12 frames a systematic paradigm-level comparison between ODI and SDI.Source: Table 12 · p. 14

9.2 Historical Positioning

  • The paper characterizes progress from Tikhonov to Adam as development within one paradigm, and the move from ODI to SDI as a shift across paradigms.Source: §9.2 · p. 14

9.3 As a Paradigm Extension

  • SDI is presented as complementing ODI’s shortcomings in efficiency, decoupling, and interpretability, while extending the focus toward co-design of structure and algorithm.Source: §9.3 · pp. 14–15, Table 13 · p. 15

10 Discussion: Structure-Informed Methods and the Boundary of SDI

Structure-informed methods can redesign optimization objectives around problem structure, but remain optimization-driven when they do not construct a direct inverse operator. SDI and ODI are complementary: SDI offers structure-customized construction when structure is identifiable, while ODI remains the general framework when it is missing or incomplete.

10 Discussion: Structure-Informed Methods and the Boundary of SDI

  • SFDDPM-TV uses seismic structure and TV regularization, while JiT [41] predicts clean images under a manifold assumption.SFDDPM-TV fits signals using structural simplicity and spatial regularity; JiT predicts clean images rather than noise or noised quantities.Source: §10 · p. 15
  • Both methods remain optimization-driven because training and sampling still use iterative optimization and a diffusion Markov chain.Their structural priors redesign the objective rather than construct a direct inverse operator.Source: §10 · p. 15
  • ODI can incorporate structural priors, while SDI uses structure to construct a solution method for problems with identifiable structure.The paradigms are complementary: ODI remains the universal framework when structure is missing or incomplete.Source: §10 · p. 15

11 Conclusion

The conclusion presents SDI as a complement to ODI, built around mathematical and physical structure and construction of tailored solution methods. It identifies division or inversion as SDI’s core operation, while allowing structural projection to take iterative forms when needed.

11 Conclusion

  • SDI complements and extends ODI: both pursue data fitting, but ODI primarily searches iteratively while SDI primarily constructs solutions from structure.The paper frames this shift as an evolution from objective-function optimization to structure identification and a customized solution method.Source: §11 · p. 16
  • SDI has mathematical and physical driving types, with MSDPII, PSDWII, and PSDBP as current representative methods.The framework is open to additional methods as new structures are identified.Source: §11 · p. 16
  • SDI’s core operation is division, inversion, or deconvolution, contrasting with ODI’s cross-correlation or multiplication.The conclusion says deconvolution naturally decouples contributions, avoiding ODI’s mathematical coupling and eliminating the need for a Hessian matrix.Source: §11 · p. 16
  • SDI may iterate when structure requires it, with each step acting as a structural projection rather than a blind search.Whether and how to iterate depends on the nature and completeness of the available structure.Source: §11 · p. 16
  • SDI applies when a problem has identifiable mathematical or physical structure; without it, the method degenerates to ODI.The conclusion presents the paradigms as complementary parts of a methodological system for inverse problems.Source: §11 · p. 16
  • The conclusion casts SDI as a customized direct strategy that seeks to exploit intrinsic system laws, alongside ODI’s universal search strategy.The paper characterizes this as both an efficiency improvement and a deepening of cognition.Source: §11 · p. 16
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