Non-Linear Structural Probe Overview

• Non-linear structural probing is a set of advanced methods that extract intricate structural data from systems using non-linear excitations and modeling.
• These techniques leverage polynomial expansions, kernel methods, and local linearization to reveal hidden structural relationships in mechanical, fluid, and computational domains.
• Applications range from nanoscale force mapping in AFM to extracting syntactic encodings in neural networks, offering enhanced insights over linear approaches.

A non-linear structural probe is a class of methods and instrumentation designed to extract local or global structural information from systems exhibiting nonlinearity through the application of non-linear excitations, non-linear modeling, or non-linear transformations of measured data. The concept encompasses diverse application domains, including experimental condensed matter, structural mechanics, nonlinear system identification, and probing the internal representations of neural networks. Techniques denoted as non-linear structural probes are unified by their exploitation of nonlinearity—either intrinsic to the system or imposed via model design—to reveal structural relationships that are inaccessible or poorly resolved with purely linear approaches.

1. Mathematical and Theoretical Foundations

Non-linear structural probing blends the foundational ideas of linear probing—such as small-signal linearization, eigenmode extraction, and metric learning—with mathematical frameworks that directly capture or exploit higher-order interactions, nontrivial nonlinearities, intermodulation, or energy landscapes. Several rigorous mathematical formalisms underlie non-linear structural probes:

• Polynomial and Harmonic Expansions: In system identification or force spectroscopy, nonlinearity is often modeled by polynomial expansions (e.g., $$V_{\rm out}(t) = a_1 f_{\rm in}(t) + a_2 [f_{\rm in}(t)]^2 + a_3 [f_{\rm in}(t)]^3 + \cdots$$), with intermodulation terms emerging at frequencies that are integral linear combinations of the drive tones, $\omega_{m,n} = m\omega_1 + n\omega_2$ (Tholen et al., 2010).
• Kernel Methods: In probing representational structure in neural networks, kernelization of metric learning frameworks yields non-linear structural probes. For example, the RBF kernel, $$\kappa_{\mathrm{rbf}}(x,y) = \exp(-\|x-y\|^2/2\sigma^2)$$, induces a curved metric that exposes nonlinear encodings of syntactic relationships (White et al., 2021, Pal et al., 2024).
• Nonlinear System Linearization: In block-oriented systems, local linear approximations (Best Linear Approximation, BLA) at different bias points provide information on how system poles and zeros move with input, serving as a structural fingerprint of the underlying nonlinearity (Schoukens et al., 2018).
• Energy and Force Landscapes: In probing mechanical structures for buckling, the control of displacement/force reveals bifurcations and energy barriers, with key diagnostics (e.g., vertical tangents $dF/dq=0$ correspond to saddle-node bifurcations) extracted from the $F(q)$ relationship (Thompson et al., 2017).

2. Characteristic Methodologies

Non-linear structural probes utilize a rich array of methodologies designed to engage, measure, and interpret the nonlinear response of the target system.

• Multi-tone and Intermodulation Spectroscopy: Nonlinear systems, when subject to two or more drive frequencies, exhibit intermodulation products. By measuring the amplitude and phase of these products, one can reconstruct the coefficients of the nonlinear force law and, in applications such as intermodulation AFM, resolve local mechanical properties at the nanoscale (Tholen et al., 2010).
• Local Linearization & Pole-Zero Tracking: Linear approximations at various setpoints, followed by pole-zero trajectory analysis, permit structural discrimination of candidate nonlinear architectures (e.g., Wiener, Hammerstein, Wiener-Hammerstein, parallel, feedback, LFR). The movement (or immobility) of poles and zeros across operating conditions constrains the admissible system structures (Schoukens et al., 2018).
• Automated Multimodal Experiments & Active Learning: For phenomena such as nonlinear electromechanical response in ferroelectrics, automated multimodal imaging combined with local spectroscopy and machine learning—specifically deep kernel learning (DKL)—enables the disentangling of causal structural determinants of nonlinearity. Alternative hypotheses are tested via DKL, with model evidence favoring structure-function links (Liu et al., 2022).
• Microrheological Fluctuation Analysis: Driven probes in nonequilibrium fluids manifest anomalous variance scaling of position, revealing microstructural transitions in complex fluids. The breakdown of equipartition is quantified using $$\Delta_x(v) = \kappa \langle \delta x^2 \rangle / (k_B T) - 1$$, with distinct scaling regimes ($\sim v^2$ for diffusive, $\sim v^{1/3}$ for jump-dynamics) mapping the onset of substructural rearrangement (Forastiere et al., 2024).
• Kernelized Metric Learning for Neural Representations: Non-linear structural probes in neural model analysis replace linear projections with pointwise or kernel-based non-linear maps (e.g., RBF, polynomial, or sigmoid) to measure representational geometry, enabling the recovery of syntactic and structural information that is encoded non-linearly (White et al., 2021, Pal et al., 2024).

3. Instrumentation and Implementation

Instrumentation architectures for non-linear structural probes are highly domain specific but share core features designed for resolving nonlinearity:

• Lock-In Analysis on FPGAs: In intermodulation spectroscopy, field-programmable gate arrays (FPGAs) drive the device under test with multiple pure tones and perform parallel lock-in detection to extract up to 32 intermodulation components in real time. The in-phase and quadrature signals [$X_k, Y_k$], as well as amplitude and phase, are computed for real-time feedback and high-bandwidth analysis (Tholen et al., 2010).
• Active Feedback and Multi-Input Control: Structural buckling probes employ displacement- or force-controlled actuators with high-resolution load cells and encoders, with additional secondary probes or active feedback loops introduced to manage bifurcations and stabilize symmetry-breaking paths (Thompson et al., 2017).
• Optical Tweezers and Fluctuation Measurements: In microrheology, micron-scale beads are optically trapped, and their position variance under controlled drag reveals the onset of non-linear structural response of the surrounding medium (Forastiere et al., 2024).
• Structured Data Acquisition in Automated Experiments: High-throughput piezoresponse force microscopy (PFM) combines multimodal imaging (topography, amplitude, phase, resonance) with site-resolved spectral sweeps and deep kernel learning-driven acquisition planning, suited for mapping spatial heterogeneity in nonlinear coefficients (Liu et al., 2022).

4. Domain-Specific Applications

Non-linear structural probing finds direct application across several physical and engineering domains:

Application Domain	Key Non-linear Probe Method	Structural Information Extracted
Nanomechanics (AFM, PFM)	Intermodulation, BEAM, DKL	Local force laws, mechanical property maps
Structural Buckling	Lateral/axial controlled displacement	Energy barrier, instability threshold, force landscape
Fluid Microrheology	Driven probe variance scaling	Microstructural transitions, stress relaxation modes
Block-Oriented System ID	Pole/zero mobility under bias	Model class discrimination (Wiener, LFR, etc.)
Neural Representation Probing	Kernelized structural probes	Syntactic/structural encoding in embeddings
Remote Structural Monitoring	Non-imaging laser vibrometry	Intermodal coupling, transition probabilities

For example, in ferroelectric thin films, nonlinear structural probes employing PFM and DKL establish that domain walls, rather than topography, dominate the local quadratic and cubic nonlinearities (Liu et al., 2022). In synthetic and biological fluids, the scaling form of probe-position variance under nonequilibrium drive distinguishes between diffusive and activated rearrangement regimes, with the crossover mapping the onset of microstructural stress-release events (Forastiere et al., 2024).

5. Comparative Evaluation and Empirical Findings

Non-linear structural probes consistently outperform linear-only approaches when the structure of interest is nonlinearly encoded or when system response is driven far enough from equilibrium.

• Improved Syntactic Recovery in LLMs: In unsupervised structural probing of BERT representations, kernelized (RBF) probes achieve statistically significant improvements in Unlabeled Undirected Attachment Score (UUAS) and often in Spearman correlation (DSpr), compared to linear or polynomial alternatives across six languages. For example, Tamil UUAS increased from 48.52% (linear) to 56.96% (RBF) (White et al., 2021).
• Resolution of Nonlinear Force Laws at the Nanoscale: The amplitude and phase pattern across intermodulation products yields a multidimensional contrast mechanism for mapping mechanical properties and polymer force–distance laws, unattainable via classical single-frequency AFM (Tholen et al., 2010).
• Structural Discrimination in System ID: Observing the mobility patterns of poles and zeros via local linearization enables ruling out large classes of block-oriented models before full nonlinear system identification, saving modeling effort while providing structural guarantees (Schoukens et al., 2018).
• Causality Disambiguation in Multimodal Imaging: Deep kernel learning establishes Bayesian model comparison among alternative structural hypotheses, providing a principled route to identifying causal mechanisms for nonlinear material response (e.g., distinguishing domain wall-dominated from topography-induced nonlinearity) (Liu et al., 2022).

6. Limitations and Challenges

Non-linear structural probing is subject to structural, experimental, and interpretive limitations:

• Excitation Regime: Small-signal approximations fail for non-differentiable nonlinearities. Large-signal and highly nonlinear regimes demand either direct measurement of higher harmonics or model-based kernelization, which may increase estimator variance or demand more extensive data (Schoukens et al., 2018, Tholen et al., 2010).
• Instrumentation Complexity: Parallel lock-in detection, active feedback, and high-resolution multi-modal imaging require nontrivial hardware and control architectures, particularly for nanoscale or real-time applications (Tholen et al., 2010, Thompson et al., 2017).
• Interpretation Ambiguity: Mobility of poles and zeros is a necessary but not sufficient criterion for structure identification; additional model selection steps are required (Schoukens et al., 2018). Kernelized probe improvements must be distinguished from overfitting or inappropriate regularization (White et al., 2021).
• Causality Assessment: Separating causal from correlative contributors in high-dimensional data (multimodal imaging) depends on the adequacy of exploration strategies and the clarity of model evidence superiority (Liu et al., 2022).

7. Perspectives and Generalization

The framework of non-linear structural probing continues to expand in scope. Emerging directions encompass:

• Integration with Active Learning: Real-time experimental design, driven by statistical models such as deep kernel learning, enables adaptive targeting of regions most informative for unraveling nonlinear structural heterogeneity (Liu et al., 2022).
• Generalization to Multimodal and Multi-scale Systems: Techniques extend naturally to spectroscopic, chemical, and electron imaging modalities, with the principle that the structural descriptor $X$ and functional response $y$ may represent any structurally and spatially resolved data, respectively (Liu et al., 2022).
• Unified Treatment Across Physical, Mechanical, and Computational Domains: The theoretical underpinnings of non-linear structural probes—polynomial expansions, kernel metrics, energy landscapes—offer a universal language that unifies approaches to model extraction, structure discrimination, and representation probing.

Non-linear structural probes thus constitute a diverse and analytically robust toolkit for revealing and quantifying structural information in systems where linear approaches are inadequate, leveraging the information content inherent in nonlinear response, higher-order coupling, or kernel-induced geometries. Their domain-adaptivity and increasing integration with statistical learning architectures support their ongoing adoption in both experimental and computational research settings.