Data-Free Model Order Reduction Techniques

• Data-Free Model Order Reduction is a technique that constructs reduced-order models using analytic structures and operator properties instead of large snapshot datasets.
• The method employs Taylor expansions and recursive Fréchet derivatives to capture parameter sensitivities and reduce computational overhead.
• Applications in seismic inversion, parametric PDEs, and optimization demonstrate its efficiency and robustness in data-constrained scenarios.

Data-Free Model Order Reduction (MOR) refers to a class of techniques for constructing reduced-order models (ROMs) that do not rely on extensive data generated from simulation or experiment across a wide parameter space. Instead, these methods leverage analytic structure, operator properties, or targeted sampling to efficiently approximate the behavior of large-scale dynamical or parametric systems. Data-free MOR approaches avoid the prohibitive computational overhead of generating and storing high-fidelity snapshots at many parameter values, and frequently apply when access to system internals is limited or where computational resources are constrained.

1. Core Principles of Data-Free Model Order Reduction

Data-free MOR methods construct low-dimensional surrogate models by utilizing a priori analytical knowledge, operator-based projections, or on-the-fly basis generation, rather than amassing a large database of full-order solutions. The defining attributes include:

• Avoidance of global snapshot sets spanning parameter or input space (in contrast to standard POD or RB approaches).
• Direct use of mathematical models, operator affinities, or derivatives (e.g., Fréchet derivatives, transfer function evaluations).
• Emphasis on “black-box” reduction procedures, which do not require internal access to all system matrices (relying instead on transfer function, system response, or derivative information).
• Efficient handling of parameter variations, especially when solutions depend smoothly on parameters.

This paradigm enables scalable reduction in, for example, inverse problems, parametric PDE-constrained optimization, or settings with expensive forward solvers, such as seismic wave propagation (Besset et al., 27 Mar 2025), nonlinear finite element analysis (Li et al., 16 Sep 2024), or nuclear physics emulators (Melendez et al., 2022).

2. Taylor Expansion and Fréchet Derivative-Based Bases

A key innovation exemplified by the MOR-Tₗ method is the exploitation of Taylor polynomial expansions and Fréchet derivatives to construct ROM bases that are robust with respect to parameter changes (Besset et al., 27 Mar 2025). Rather than constructing the reduced basis solely from the full-order solution at a reference parameter value θ₀, the method augments the basis with a sequence of Fréchet derivatives:

$$P_{\theta_0}(u_\theta) = \sum_{\ell=0}^L \frac{1}{\ell!} v_{\theta_0}^\ell$$

where $v_{\theta_0}^\ell$ is the ℓth Fréchet derivative of the solution $u_\theta$ with respect to the parameter at θ₀.

Each derivative vector is computed by solving a well-defined boundary value problem with a recursively defined right-hand side:

$$f^{(\ell)} = -\sum_{k=0}^{\ell-1} \binom{\ell}{k} \left(D^{(\ell-k)} \mathcal{L}_{\theta_0}\right)(v_{\theta_0}^k)$$

This construction provides sensitivity information so that the reduced basis “bakes in” how the solution deforms under parameter perturbations. The approach is “data-free” in the sense that no sampling over the parameter space is required; only a limited set of augmented solves at a single reference configuration suffices.

3. Multiple-Right-Hand-Side (RHS) Strategy

The recursion for Fréchet derivatives naturally leads to a hierarchy of linear systems sharing the same operator but differing in their right-hand sides. MOR-Tₗ leverages this by solving for all derivatives (up to degree L) in a block fashion. The computational process is as follows:

1. Solve the full-order problem at θ₀ (unperturbed).
2. For each ℓ > 0, solve a PDE with right-hand side f^{(ℓ)}, which is constructed recursively using lower-order derivatives.
3. Aggregate all solution and derivative snapshots into a single matrix.

Because each solve involves the same or nearly the same linear operator, efficient linear algebraic solvers (including block or parallel implementations) can handle all right-hand sides simultaneously. Once the snapshots are collected:

• A secondary reduction step (QR or SVD) compresses the set, yielding basis functions φ_i for the ROM.

This strategy substantially minimizes extra computational effort; the number of full-size solves required is only O(L), and these can often be accelerated collectively.

4. Computational Efficiency and Parameter Robustness

By explicitly capturing parameter sensitivities, data-free approaches such as MOR-Tₗ exhibit:

• Offline–online decomposition: The relatively expensive generation of derivative-based snapshots is contained in a one-time offline stage.
• Low online cost: ROM time integration and parameter updates are conducted in a space of dimension N ≪ original system size.
• Robustness to parameter perturbations: ROMs constructed with derivative-augmented bases maintain accuracy under moderate parameter variations (up to 20% perturbation in velocity for seismic wave inversion (Besset et al., 27 Mar 2025)).
• Improved convergence in inversion: In tasks such as seismic full waveform inversion, the enhanced basis supports efficient line searches and updates even in the presence of noise.

Numerical experiments confirm high-fidelity approximations with error well below 10⁻²% in benchmark wave problems, and up to 4× faster online evaluations compared to traditional SVD-based POD approaches.

5. Applications in Seismic Wave Propagation and Inverse Problems

MOR-Tₗ and related data-free MOR concepts are particularly suited to parametric PDE settings where repeated simulations are necessary across different model configurations. In seismic inversion, the acoustic wave equation depends on spatially distributed velocity parameters θ(x):

$θ \, \frac{∂^2 u_θ}{∂ t^2} - Δ u_θ = f(x, t)$

By precomputing both the base solution and its Fréchet derivatives at a reference velocity, the ROM can accurately and efficiently predict wavefields for new θ close to θ₀. In practical inversion workflows, such as full waveform inversion (FWI), this capability accelerates each iteration of parameter update optimization. The approach is similarly extendable to other parametric inversion problems, given differentiability with respect to parameters.

6. Broader Relevance and Suitability for Data-Free Scenarios

The principal advantages of data-free MOR are realized when traditional snapshot-based reduced basis approaches are infeasible due to cost, storage, or system access limitations. The following features make data-free MOR uniquely suitable in such environments:

• Analytic or operator-based basis construction fully decouples reduction from exhaustive data generation.
• The methodology is naturally compatible with black-box solvers, as it does not require access to detailed system matrices at multiple parameter values.
• The multiple-RHS and derivative-based construction can be implemented using existing high-fidelity solvers with minimal modification.
• The approach aligns with broader trends in computational science, where large-scale parametric studies and optimization routines require vast numbers of forward solves, and where data storage and transfer bottlenecks must be avoided.

Summary of central mathematical expressions:

Concept	Formula	Section
Taylor expansion of solution	$$P_{\theta_0}(u_\theta) = \sum_{\ell=0}^L \frac{1}{\ell!} v_{\theta_0}^\ell$$	2, 6
Recursive RHS for derivatives	$$f^{(\ell)} = -\sum_{k=0}^{\ell-1} \binom{\ell}{k} (D^{(\ell-k)} \mathcal{L}_{\theta_0})(v_{\theta_0}^k)$$	3
Reduced solution approximation	$$u_{\theta_a}(t) \approx \sum_{i=1}^N a_i(t) \varphi_i(x)$$	3, 6

These developments collectively illustrate the state of the art in data-free model order reduction, as exemplified in the MOR-Tₗ method for seismic wave propagation and related high-dimensional parametric PDE applications (Besset et al., 27 Mar 2025).