GenICON: Probabilistic Operator Learning

• GenICON is a framework that extends the classical ICON operator mapping to generate full posterior distributions with principled uncertainty quantification in differential equations.
• It employs conditional generative architectures, such as conditional GANs, to construct a latent-to-solution mapping that captures posterior variability from contextual data.
• The approach integrates Bayesian inference into operator learning, enabling rigorous uncertainty management in forward and inverse problems across ODEs and PDEs.

The generative formulation of ICON (GenICON) is a probabilistic operator learning paradigm in which the classical ICON architecture, built for mapping sets of differential equation conditions (initial/boundary data) to solution operators, is extended to yield full posterior predictive distributions rather than solely point estimates. This enables principled uncertainty quantification, providing both sample diversity and probabilistic confidence in solution predictions for operator learning in ordinary and partial differential equations.

1. Probabilistic Operator Learning Framework

ICON is framed within the context of random differential equations (RDEs), modeling the solution operators as distributions over function spaces (typically Banach or Hilbert). Training data comprises tuples of parameters $\alpha$, conditions $y$, and quantities of interest $z$, drawn as

$$P_{(\alpha, y, z)} = P_\alpha \otimes P_y \otimes P_{z|y, \alpha}$$

Critically, only the marginal over $(y, z)$ is accessible during training, each demonstration set corresponding to a fixed (unobserved) $\alpha$. This context structure enables ICON to utilize example condition-solution pairs for implicit parameterization of the solution operator.

In ICON, the prediction for an unseen condition $y^J$ given context demonstrations $\{(y^j, z^j)\}_{j=1}^{J-1}$ is the Bayesian posterior predictive mean

$$T^*(y^J; \{(y^j, z^j)\}) = \mathbb{E}[z^J | y^J, \{(y^j, z^j)\}]$$

where the posterior predictive distribution is

$$Q_{z^J|y^J,\{(y^j,z^j)\}}(\cdot) = \int P_{z^J|y^J, \alpha}(\cdot | y^J, \alpha) \cdot P_{\alpha | \{(y^j, z^j)\}}(d\alpha)$$

This implicitly realizes Bayesian inference over operators conditioned on observed context.

2. GenICON: Generative Extension via Posterior Sampling

GenICON advances ICON by constructing a generative mapping $G$ from a latent space (e.g., Gaussian) into solution samples, enabling generation of multiple solutions reflecting posterior uncertainty:

$G : H \times Y \times (Y \times Z)^{J-1} \to Z$

For random $x \sim P_\eta$ in $H$ and context $\{(y^j, z^j)\}$, the pushforward

$$G(\cdot, y^J, \{(y^j,z^j)\})_\# P_\eta = Q_{z^J | y^J, \{(y^j,z^j)\}}$$

Samples from $G$ recover the full distribution of plausible solutions, not just the mean. The expectation over generated samples yields the classical ICON output:

$$\int G(x, y^J, \{(y^j,z^j)\}) P_x(dx) = T^*(y^J; \{(y^j,z^j)\})$$

Practically, GenICON is instantiated via conditional generative architectures such as conditional GANs, trained to align the joint density of generated pairs $(y^j, z^j)$ with empirical distributions; divergence minimization (e.g., forward KL with Lipschitz regularization) can be used for model fitting.

3. Mathematical Formulation and Example

For training,

$$\min_\theta \frac{1}{M} \sum_{m} \left\| z^J_m - T_\theta(y^J_m; \{(y^j_m, z^j_m)\}_{j=1}^{J-1}) \right\|_Z^2$$

where $T_\theta$ is the ICON transformer, learned to perform regression against withheld targets $z^J_m$ over demonstration contexts.

Consider, for instance, an RDE

$$\frac{d}{dt} u(t, \omega) = \gamma_1(\omega) c(t, \omega) u(t, \omega) + \gamma_2(\omega), \quad u(0, \omega) = u_0(\omega)$$

The "condition" is $(c(t, \omega), u_0(\omega))$, the QoI is $u(t, \omega)$. Demonstrations given to ICON are pairs of condition/signal, and GenICON—upon receiving new $c^J$, $u_0^J$, and a context—can produce distributional samples for $u^J(t, \omega)$ reflecting epistemic uncertainty due to both model ambiguity and data variability.

4. Applications and Implications

The generative formulation is suited to scenarios with model uncertainty, non-identifiability, or noisy/incomplete data in operator learning. Example domains include:

• Forward/inverse problems in PDE and ODE modeling where the space of admissible operators is not deterministic.
• Uncertainty quantification for scientific computing where construction of credible intervals and risk metrics from solution samples is essential.
• Inverse problems where multiple parameterizations fit observed data, allowing GenICON to reflect the distribution over possible operators.

The probabilistic formalism of GenICON integrates Bayesian methodology into foundation model architectures for differential equations, enabling robust, sample-efficient operator learning and rigorous propagation of uncertainty.

5. Broader Significance and Future Directions

By casting operator learning within a random differential equation and Bayesian framework, GenICON bridges deterministic operator regression with flexible generative modeling. The capacity to perform conditional posterior predictive sampling elevates ICON from deterministic mapping to a stochastic process, facilitating uncertainty-aware predictions for a wide class of data-driven scientific computing applications.

Further research directions include:

• Integration with advanced generative modeling paradigms (Gaussian processes, stochastic normalizing flows).
• Extension to high-dimensional settings, inverse design workflows, and multi-modal operator tasks.
• Development of conditional generative architectures and scalable divergence minimization strategies tailored to large operator datasets.

This synthesis—a mathematical formalism leveraging context-driven Bayesian inference and generative sampling—provides an extensible basis for future advances in probabilistic operator learning.