Mathematical Artificial Data Framework

• The Mathematical Artificial Data (MAD) framework is a physics-embedded paradigm that leverages analytical solutions to generate precise synthetic data for learning differential equation operators.
• It employs analytical decomposition, Green's functions, and trigonometric methods to create error-free, high-fidelity datasets across diverse PDE problems.
• MAD enhances operator learning by offering rapid data generation, superior accuracy, and broad compatibility with various neural architectures in scientific computing.

The Mathematical Artificial Data (MAD) Framework is a physics-embedded, data-driven paradigm for operator learning, designed to efficiently generate high-fidelity synthetic data for the training of machine learning models that approximate solution operators to differential equations. By exploiting the mathematical structure of underlying physical systems, MAD eliminates the requirement for costly experimental or simulated data, thereby achieving efficient and accurate operator learning across diverse multi-parameter problems in scientific computing (2507.06752).

1. Foundational Principles

MAD is introduced as a universal approach for operator learning problems, grounded in the observation that solutions to differential equations (DEs) can often be expressed analytically or semi-analytically by leveraging the intrinsic properties of the governing equations. MAD constructs synthetic datasets by analytically solving or decomposing DEs for randomly selected (but mathematically well-posed) boundary conditions and source terms, ensuring that all generated data satisfy the relevant physical laws up to machine precision. This results in error-free or near-error-free input–output pairs, which can be used as a training set for neural operator approximators.

The central objective of the framework is to learn operators

$\phi: (f, g) \mapsto u,$

where $f$ denotes the source function, $g$ the boundary condition, and $u$ the solution to a given DE on domain $\Omega$ with boundary $\partial\Omega$.

2. Physics-Embedded Data Generation

MAD distinguishes itself from traditional, simulation-based or experimental data pipelines by employing direct mathematical constructions for data generation. The primary workflow is as follows:

1. Analytical Decomposition: For a PDE such as

$$\mathcal{L}u = f \text{ in } \Omega, \quad u = g \text{ on } \partial\Omega$$

the solution is decomposed as $u(x) = u_1(x) + u_2(x)$, where: - $u_1$: solves the homogeneous equation ($f=0$), nontrivial boundary ($g\neq 0$), - $u_2$: solves the inhomogeneous equation but with homogeneous boundary ($g=0$).

1. Synthetic Solution Generation: For each component, a distinct method is used:
   • Source-containing problems (MAD0): Utilize neural networks with sine activations, e.g., architectures like $[2, 50, 50, 1]$, capable of representing oscillatory or highly regular solutions.
   • Source-free cases: Two principal approaches are defined:
     • Fundamental solution-based (MAD1): Analytical solutions via Green's functions, such as $(1/2\pi) \ln \|\mathbf{x}-\mathbf{y}\|$ for 2D Laplace, or $-1/(4\pi\|\mathbf{x}-\mathbf{y}\|)$ for 3D Laplace equations. For the Helmholtz equation, combinations of Bessel functions $J_0$ and $Y_0$ are prescribed.
     • Trigonometric/hyperbolic function-based (MAD2): Linear combinations of functions like $$(A_i \cos(a_i x) + B_i \sin(a_i x))(C_i \cosh(a_i y) + D_i \sinh(a_i y))$$ are sampled, with coefficients chosen from standard distributions, ensuring diversity and boundary adherence.
   • Boundary and Source Term Sampling: Boundary profiles $g$ are drawn from Gaussian Random Fields, typically characterized by a Radial Basis Function (RBF) kernel,

   $$k_l(x_1, x_2) = \exp\left(-\frac{|x_1-x_2|^2}{2l^2}\right)$$

   and source terms $f$ are generated and smoothed to ensure physical plausibility.

This data generation is agnostic to any particular network architecture, producing arbitrarily large datasets with precise mathematical control over input distributions and solution regularity.

3. Integration with Machine Learning Architectures

MAD-generated datasets are compatible with a variety of operator learning architectures, including (but not limited to):

• DeepONet: For mapping pairs of functions $(f, g)$ to solution fields.
• Dual-branch networks: Separate encoders process $g$ and $f$, whose latent representations are combined to yield $u$, supporting modular and interpretable operator decomposition.
• Fourier Neural Operators (FNO): Benefiting from diverse and accurate function samples spanning the solution space.
• Other scientific neural architectures: The modular nature of MAD allows deployment in hybrid, multitask, or transfer learning workflows within scientific computing.

Supervised learning is typically performed using a mean squared error (MSE) objective due to the high accuracy of the input–output labels.

4. Performance, Efficiency, and Accuracy

The efficiency and accuracy advantages afforded by the MAD framework arise from several sources:

• Rapid Data Generation: Analytical and semi-analytical methods yield training samples orders of magnitude faster than finite-difference or finite-element solvers. For instance, 200 samples for a 2D Helmholtz problem are produced in approximately 2.19 seconds, versus thousands of seconds for conventional solvers.
• Error-Free Labels: Training data are exact solutions to the governing equations; thus, learning is not impaired by label noise or discretization error, in contrast to simulation-derived datasets.
• Superior Convergence: Experiments demonstrate lower training and validation loss, as well as smaller relative $L^2$ errors, for operator approximators trained under the MAD regime compared to Physics-Informed Neural Networks (PINNs) or simulation-driven methods.
• Generalizability: Because MAD decouples function sampling from neural architecture, it supports varied geometries (square, disk, L-shaped domains), higher-dimensional PDEs, and a wider parameter range (e.g., variable wavenumbers in Helmholtz problems).

A comparison table summarizing these advantages, as reported, is as follows:

Approach	Data Generation Time (2D Helmholtz)	Training Data Accuracy	Generalizability
MAD (all variants)	~2 seconds (200 samples)	Analytical / error-free	High—arbitrary domains
Numerical Solvers	10³–10⁴ seconds (same samples)	Discretization/round-off	Moderate
PINN	N/A (learns directly from PDE)	Sensitive to hyperparameters	Lower (harder to scale)

5. Applications and Demonstrations

MAD has been applied to a range of canonical PDEs relevant in scientific computing:

• Poisson equation: $\nabla^2 u = f$, $u = g$ on $\partial\Omega$.
• Helmholtz equation: $\nabla^2 u + k u = f$, $u = g$ on $\partial\Omega$, with parametric dependence on the wavenumber $k$.
• Laplace equation: $\nabla^2 u = 0$, source-free settings.

The framework addresses general operator approximation tasks, such as

$$\phi: (f, g) \mapsto u, \qquad \phi_1: g \mapsto u.$$

Numerical demonstrations cover 2D domains (squares, disks, L-shaped) and extend to 3D and complex-geometry settings, illustrating robustness and scalability.

6. Implications and Future Prospects

MAD represents a foundational advance for physics-informed machine intelligence:

• Universal Operator Learning: Its physical rigor and modularity position it as a candidate for a universal operator learning paradigm.
• Hybrid Integration: MAD-generated data can be used to augment or constrain neural PDE solvers, such as PINNs, or as part of transfer learning pipelines.
• Scalable Scientific Computing: By sidestepping simulation bottlenecks, MAD may allow real-time and large-scale deployment of learned models in engineering, multi-physics simulation, and uncertainty quantification workflows.

A plausible implication is that future operator learning research will standardize the use of physics-embedded artificial data as a complement or replacement for simulation datasets, especially as analytic techniques expand to cover more classes of boundary value problems.

7. Limitations and Outlook

While MAD eliminates many traditional bottlenecks, its performance is inherently contingent on the availability of analytical or semi-analytical expressions for the relevant DEs. In highly nonlinear, high-dimensional, or chaotic systems where such formulas are unavailable, MAD must either approximate solutions numerically or integrate with traditional surrogate modeling. Nonetheless, where analytic approaches are tractable, MAD provides a highly efficient and accurate alternative for data-driven scientific machine learning.

In summary, the Mathematical Artificial Data (MAD) Framework introduces a physics-consistent, modular method for artificial data generation in operator learning—offering unmatched label accuracy, computational efficiency, and architectural flexibility. It provides a versatile foundation for both foundational research and the practical deployment of machine learning in scientific computing (2507.06752).