PDE-Constrained Learning Framework

• Partial differential equation-constrained learning is a method that fuses physics-based PDE models with probabilistic machine learning to enable data-efficient and robust parameter estimation.
• It embeds time-dependent, nonlinear PDE operators directly into model architectures, using discretization and GP priors to enforce physical laws through principled regularization.
• The approach excels in handling sparse and noisy data across diverse systems, reliably discovering complex dynamics and nonlinear behaviors with minimal observations.

A partial differential equation (PDE)-constrained learning framework is a class of methodologies that fuses machine learning—particularly probabilistic inference and neural approximation—with the explicit structure defined by time-dependent and nonlinear PDEs. This synthesis enables data-driven discovery, identification, or prediction in scientific systems where the governing laws are only partially known or where available measurements are sparse. Such frameworks are distinguished by their integration of the underlying “physics” directly into the learning process, conferring data efficiency, robustness to noise, and principled regularization.

1. Formulation: Embedding PDEs in Machine Learning Models

The cornerstone of the PDE-constrained learning paradigm is the direct incorporation of the governing PDE operator into the statistical or machine learning architecture. For example, equations of the form

$u_t + \mathcal{N}_x^\lambda u = 0$

(where $\mathcal{N}_x^\lambda$ denotes a parametrized nonlinear spatial differential operator) are discretized in time (e.g., via backward Euler) and recast so that the relation

$\mathcal{L}_x^\lambda u^n = u^{n-1}$

serves as a prior constraint on the model’s latent field at time step $n$. In the “hidden physics” Gaussian process (GP) model (Raissi et al., 2017), a multi-output GP prior $u^n(x) \sim \mathcal{GP}(0, k(x, x'; \theta))$ is imposed, and the corresponding covariance structure is defined by applying the relevant differential operator to the kernel. This ensures that physical knowledge—such as conservation laws or dynamics encoded by the PDE—directly shapes the architecture’s prior or regularization.

2. Data Efficiency and Regularization

A defining property of PDE-constrained frameworks is their ability to remain data-efficient, exploiting the encoded prior knowledge to reduce sample complexity. By embedding the PDE structure within the kernel or architecture, these models often require only minimal, scattered observations (sometimes just two time snapshots or tens of points, where fully data-driven methods need thousands). The learning objective is almost always grounded in marginal likelihood, with regularization provided not by uninformative penalties or hand-tuned hyperparameters, but by the log-determinant term: $$- \log p(h | \theta, \lambda, \sigma^2) = \frac{1}{2} h^\top K^{-1} h + \frac{1}{2} \log|K| + \frac{N}{2} \log(2\pi)$$ where model complexity and data fit are automatically balanced—an Occam’s razor principle that discourages overfitting under small-data regimes.

3. Canonical Applications and Demonstrated Results

These frameworks have been applied to a broad spectrum of physically and mathematically rich PDEs, including:

Equation Type	Example PDE	Key Capability Demonstrated
Burgers’ Equation	$u_t + \lambda_1 u u_x - \lambda_2 u_{xx} = 0$	Recovery of nonlinear advection–diffusion dynamics from two sparse time snapshots
Kuramoto–Sivashinsky	$$u_t + \lambda_1 u u_x + \lambda_2 u_{xx} + \lambda_3 u_{xxxx} = 0$$	Correct identification of stabilizing and destabilizing higher-order terms
Nonlinear Schrödinger	$i h_t + \lambda_1 h_{xx} + \lambda_2 |h|^2 h = 0$	Handling coupled real/imaginary fields with minimal observations
Navier–Stokes (2D, incompressible)	System as in $$u_t + \lambda_1 (u u_x + v u_y) = -p_x + \lambda_2 (u_{xx} + u_{yy})$$	Parameter estimation in multi-field fluid systems (with the stream function constraint)
Time-dependent fractional PDE	$$u_t - \lambda_1 \mathcal{D}^{\lambda_2}_{-\infty,x} u = 0$$	Identification of fractional orders and anomalous diffusion characteristics

Parameter estimation is robust even when data is scattered or corrupted; nonlinear coefficient recovery (including for non-polynomial or fractional operators) is feasible within this paradigm.

4. Methodological Foundations

The methodology is structured as follows:

• The governing PDE is discretized (e.g., via backward Euler) and possibly linearized if nonlinear.
• A GP prior is placed on the latent field, with physical operators applied to build the joint prior structure across time steps.
• Model hyperparameters (including coefficients inside the PDE operator) are treated as kernel hyperparameters.
• Training proceeds by maximizing the log marginal likelihood, simultaneously fitting observed data and favoring parsimonious models.

This process is agnostic to the choice of observation points, requiring neither grid-aligned data nor high-resolution temporal sampling, and can adapt to complex domains through judicious formulation of the underlying kernel and operator algebra.

5. Practical Implications and Advantages over Alternatives

Direct PDE constraint embedding yields multiple practical benefits:

• Noise-robustness: The analytic application of differential operators to the covariance kernel circumvents the need for potentially unstable numerical differentiation.
• Flexibility: The approach allows for scattered, irregular, or missing data in space and/or time, broadening applicability to realistic experimental settings.
• Automatic uncertainty quantification: The GP posterior yields credible intervals and explicit confidence statements about inferred quantities.
• Descriptor discovery capability: When the operator form is partially known, the same framework can be used for discovering or isolating unknown functional forms—outperforming dictionary-based sparse regression in scenarios with fractional, transcendental, or otherwise atypical terms.

Compared to traditional regression-based PDE discovery (e.g., SINDy or related methods), the GP-based PDE-constrained framework is fundamentally more data-frugal, avoids problematic hyperparameter sweeps, and more readily generalizes across noise regimes and operator families.

6. Comparative Analysis and Limitations

Direct comparison to dictionary approaches highlights both strengths and limitations:

• Sample efficiency: Only a handful of measurements may suffice, as opposed to thousands for regression-based identification.
• Noise: GP structure naturally accommodates noisy measurements; regression methods often require pre-filtering.
• Complex operators: Approaches that encode the physics in the kernel can robustly estimate PDE parameters in settings where traditional, term-expansion dictionaries (even with cross-validation) are limited—e.g., for discovery of fractional-derivative PDEs or complex nonlinearities (e.g., $\sin(\lambda u)$).
• Limitation: GP-based frameworks may scale less directly to extremely large data sets or very high-dimensional spatial domains due to the cubic scaling in the number of observations, though this concern is mitigated by data efficiency.

7. Summary and Outlook

PDE-constrained learning frameworks, exemplified by the hidden physics GP paradigm (Raissi et al., 2017), represent a principled integration of classical mathematical physics and probabilistic machine learning. By encoding the law of evolution as a constraint directly into the probabilistic model structure, these frameworks achieve robust, data-efficient inference, accurate parameter identification, and principled uncertainty quantification across a suite of canonical and complex PDEs. Their ability to operate with minimal data, manage noise, and adapt to diverse operator forms makes them highly suitable for real-world applications where data collection is expensive and the governing equations are partially or wholly unknown. The approach is a significant evolution in the landscape of scientific machine learning, uniting model-driven priors and data-driven inference for complex dynamical systems.