Operator-Based Machine Intelligence

• Operator-based machine intelligence is a paradigm that generalizes conventional learning by synthesizing and composing operators across infinite-dimensional function spaces.
• It integrates neural, spectral, and kernel-based architectures to achieve discretization-independent, scalable learning for tasks like PDE surrogate modeling and symbolic reasoning.
• The approach fosters enhanced interpretability and rapid transfer learning through reinforcement learning and operator induction, advancing diverse applications from scientific computing to industrial automation.

Operator-based machine intelligence is a paradigm that formulates intelligence and learning tasks as the synthesis and composition of operators—mathematical or programmatic entities that act on structured representations such as functions, rules, or programs. This approach moves beyond traditional finite-dimensional models, encoding data, hypotheses, and behaviors in spaces where operators perform flexible transformations, and learning itself is cast as operator induction. The following sections survey key theoretical foundations, practical methodologies, architectural characteristics, core application domains, and current research frontiers in operator-based machine intelligence.

1. Foundations: Operators and Learning in Function Spaces

At its core, operator-based machine intelligence generalizes the learning problem from approximating functions $f: \mathbb{R}^n \to \mathbb{R}^m$ to learning operators $T: \mathcal{H}_x \to \mathcal{H}_y$ between (typically infinite-dimensional) Hilbert or Banach spaces of functions (Kiruluta et al., 27 Jul 2025, Kovachki et al., 24 Feb 2024). This formulation is instantiated in several domains:

• Learning solution operators for partial differential equations (PDEs), where $T$ maps input fields (e.g., initial conditions, parameters) to output fields (e.g., solutions) (Kovachki et al., 24 Feb 2024, Chen et al., 2023).
• Symbolic and programmatic domains, where an operator encodes possible transformations of rules, programs, or symbolic structures (Martínez-Plumed et al., 2013).
• Signal processing and representation theory, where integral or spectral operators provide translation, filtering, or invariance properties (Kiruluta et al., 27 Jul 2025).

The Hilbert space formalism enables learning tasks to be formulated as regularized empirical risk minimization over operators:

$$\min_{T \in \mathcal{B}(\mathcal{H}_x, \mathcal{H}_y)} \sum_{i} \|T f_i - g_i\|^2_{\mathcal{H}_y} + \lambda \|T\|^2_{HS}$$

where $\|T\|_{HS}$ is the Hilbert–Schmidt norm, and the finite sum typically represents observed data (Kiruluta et al., 27 Jul 2025).

A central advantage of this operator-based view is that it enables discretization-independent, scalable learning: models trained at one resolution or on one set of representations can in principle generalize across varying resolutions or domains.

2. Meta-Programming, User-Defined Operators, and Reinforcement Learning

Operator-based machine intelligence systems often elevate operators to first-class objects. In the framework of gErl (Martínez-Plumed et al., 2013), for instance, the user can specify custom operators as explicit code transformations or meta-operators that generalize over rule positions and term manipulations:

$$\mu O :: \text{Pos} \times \mathcal{T}(\Sigma, \mathcal{X}) \to \mathcal{O}$$

where $O$ is the set of operators, $\text{Pos}$ identifies locations in an abstract syntax tree, and $\mathcal{T}$ denotes terms over a signature.

Operators become the actions in a reinforcement learning (RL) loop: the system iteratively applies operators to rules, receives feedback via an optimality measure, and adapts its operator selection policy. Actions are represented as tuples $\langle o, \rho \rangle$ (operator and rule), and policy learning proceeds via function approximation over feature spaces (e.g., Q-matrix updates). Transfer learning is inherent, as learned policies over abstract operator-rule features can be transferred across tasks—facilitating quick adaptation to new, even structurally distinct, problem domains (Martínez-Plumed et al., 2013).

3. Operator Learning Architectures: Neural and Spectral Approaches

Modern operator learning architectures predominantly fall into “neural operator” families or spectral operator frameworks.

Neural Operators

Neural operators generalize neural networks to map functions to functions (Kovachki et al., 24 Feb 2024, Chen et al., 2023, Kissas et al., 2022). Canonical architectures include:

• DeepONet: Implements the operator via a finite expansion with neural networks learning data-dependent coefficients (“branch net”) and basis functions (“trunk net”):

$$\Psi_{\text{DEEP}}(u;\theta)(y) = \sum_{j=1}^d \alpha_j(L u;\theta_\text{branch}) \cdot \psi_j(y;\theta_\text{trunk})$$

• Fourier Neural Operator (FNO): Replaces standard convolutions with nonlocal Fourier-domain convolutions, enabling efficient representation and transfer across spatial resolutions:

$$L_l(v)(x) = \sigma(W_l v(x) + b_l + K(v)(x;\gamma_l))$$

with $K(v)$ as a Fourier multiplier operator.

• Radial Basis Operator Networks (RBON): Use kernel expansions with radial basis functions over both input samples and output queries, maintaining very low error even in out-of-distribution settings (Kurz et al., 6 Oct 2024):

$$G^\dagger(u^m)(\cdot) = \sum_{i=1}^M \sum_{k=1}^N \xi_i^k \cdot g(\lambda_i \|u^m - \mu_{ik}^m\|_{\mathbb{R}^m}) \cdot g(\omega_k\|\cdot - \underline{k}\|_{\mathbb{R}^d})$$

On non-Euclidean domains, the NORM architecture projects functions into the space spanned by Laplace–Beltrami eigenfunctions, enabling neural operators to address PDEs and mappings on arbitrary Riemannian manifolds (Chen et al., 2023).

Spectral and Kernel-Based Operators

Hilbert space spectral theory underpins a class of operator models based on Fourier, wavelet, scattering transforms, and reproducing kernel Hilbert spaces (RKHS) (Kiruluta et al., 27 Jul 2025). For example, spectral filtering operators act diagonally in frequency, and reasoning operations are encoded as operator composition or projection.

RKHS theory provides a representer theorem guaranteeing that, for regularized loss functions, the solution lies in the span of kernel evaluations:

$f^*(x) = \sum_i \alpha_i K(x, x_i)$

where $K$ is a reproducing kernel.

Scattering transforms stack wavelet convolutions and modulus operators, providing analytically constructed, stable, and invariant representations—competitive with trainable deep networks in certain contexts.

Koopman operators extend this approach to dynamical systems, providing linear lifted representations of generally nonlinear systems for control and prediction.

4. Applications: Structured Prediction, Scientific Computing, and Reasoning

Operator-based machine intelligence encompasses a wide array of applications:

• Scientific Computing: Neural operators (DeepONet, FNO, RBON, NORM) provide surrogate models for parametric PDEs, including Darcy flow, turbulence, heat transfer, and mechanical systems, often achieving state-of-the-art or superhuman performance (Kovachki et al., 24 Feb 2024, Chen et al., 2023, Kurz et al., 6 Oct 2024).
• Symbolic Problem Solving: Systems like gErl solve IQ test problems, list pattern detection, and cognitive tasks by evolving rules through learned operator applications and leveraging transfer of operator policy (Martínez-Plumed et al., 2013).
• Human–Robot Interaction: Operator intent inference, via Bayesian or supervised (Random Forest) methods, enables shared autonomy and dynamic teaming by recognizing operator goals in teleoperation (Panagopoulos et al., 2021, Tsagkournis et al., 2023).
• Signal Processing and Forecasting: Spectral operator methods, e.g., those employing RKHS representations or scattering transforms, extract stable, interpretable features and enable high-precision prediction of complex time series (Kiruluta et al., 27 Jul 2025, Kurz et al., 6 Oct 2024).
• Industrial Automation: Agentic, intent-driven AI orchestration architectures decompose user-provided goals into actionable operator-based tasks within collaborative industrial systems (Romero et al., 5 Jun 2025).
• Neural Operator Surrogates: In wireless communications, network management, and simulation, operator-based approaches underpin self-adaptive, automated resource allocation by integrating AI modules directly into the operating fabric (Iqbal et al., 2021, Grigorescu et al., 2 Sep 2024).

5. Learning Algorithms, Transfer, and Scalability

Learning in operator-based systems is often realized via a blend of reinforcement learning (choosing operator applications), supervised gradient descent (fitting operator coefficients), and automated optimization (auto-tuning hyperparameters or hardware primitives).

Transfer learning and multi-operator learning (MOL) frameworks allow knowledge acquired in one operator setting to support rapid adaptation in structurally novel or data-scarce situations (Martínez-Plumed et al., 2013, Zhang, 3 Apr 2024). Distributed training schemes (e.g., MODNO) separate the learning of shared input encoding and dedicated output basis functions, optimizing efficiency without sacrificing representational flexibility.

Approximation theoretical analysis shows that neural operator architectures enjoy universal approximation properties over compact sets in suitable Banach or Hilbert spaces. However, the curse of parametric complexity—an exponential dependence of parameter count on precision for general lipschitzian operators—remains a significant barrier, only sometimes mitigated for holomorphic or structured operator classes (Kovachki et al., 24 Feb 2024).

6. Interpretability, Symbolic Reasoning, and Limitations

Operator-based models offer interpretability advantages via their grounding in spectral theory, basis expansion, and explicit operator composition (Kiruluta et al., 27 Jul 2025). Symbolic reasoning can be encoded as operator sequences or as algebraic relations over Hilbert embeddings. For instance, relational reasoning is expressed as:

$T_{r_2} \circ T_{r_1} f_a \approx f_c$

mirroring multi-step logical inference.

However, performance depends critically on basis selection (Fourier, wavelet, kernel, or data-adaptive), and spectral operator models may underperform when the data's structure is mismatched to the chosen representation. Scalability to high-dimensional, multi-modal data remains an open challenge—mitigated by sparse approximations, hierarchical strategies, and GPU/TPU integration (Kiruluta et al., 27 Jul 2025).

7. Future Directions

Emerging research directions in operator-based machine intelligence include:

• Learning adaptive, data-driven operator bases that tailor spectral or functional expansions to specific domains (Kiruluta et al., 27 Jul 2025).
• Incorporating causal and temporal structure with time-varying or non-stationary operator representations, potentially unifying Koopman theory and causal inference.
• Integration of operator learning with hardware-specific optimization frameworks, allowing automated generation of tensor operators tuned for diverse architectures (Zhang et al., 8 May 2025).
• Broadening operator-based surrogates for real-world decision automation, including end-to-end reasoning frameworks that integrate semantic encoding, counterfactual analysis, and metacognitive monitoring (Wang et al., 2 Jun 2025).
• Unifying generative and reasoning capabilities via the composition of operator modules spanning symbolic, spectral, and neural domains.

This synthesis underscores operator-based machine intelligence as a principled and flexible paradigm, integrating functional analysis, deep learning, program synthesis, and reasoning to address complex learning, inference, and decision problems in both scientific and real-world settings.