In-Context Operator Networks: A Research Program for Numerical Intelligence

Intelligence is commonly understood as the ability to acquire and apply knowledge, adapt to unfamiliar situations, and solve new problems. Large language models exhibit this capacity by inferring task-relevant knowledge from textual context and applying it to new tasks. Yet intelligence need not be confined to language. Scientific and social systems often reveal themselves numerically before we can fully describe them in words: traffic moves across road networks, water flows through river basins, and physical fields evolve over space and time. These observations contain predictive relations that may be learned before they are named, formalized, or built into a specialist model. We call the ability to acquire and apply knowledge from such numerical context numerical intelligence.

Many scientific and social problems can be viewed as learning an operator that maps one function or field to another. Our work asks whether a frozen model can infer that operator directly from numerical examples, instead of requiring a separately trained model for every new system.

This is the central idea behind In-Context Operator Networks (ICON). Rather than fitting one model to one fixed operator, ICON takes a small set of input-output examples in its context and predicts the output for a new input without updating the weights at test time. In our earlier papers we used the terms condition and quantity of interest (QoI). Here we instead use key function and value function, which are closer to the broader literature.

For a concrete example, consider a 1D conservation law

\[\partial_t u(t,x) + \partial_x f(u(t,x)) = 0.\]

Its forward operator can be written as \(\mathcal{F}_{f,\tau}[u(0,\cdot)] = u(\tau,\cdot)\), or more compactly \(\mathcal{F}_{f,\tau}(u_0) = u_\tau\). Classical numerical methods approximate this operator once the governing equation is specified. In fixed-operator learning, one would train a model \(G_\theta\) for a particular choice of \(f\) and \(\tau\) so that \(G_\theta(u_0)\approx \mathcal{F}_{f,\tau}(u_0)\). If the operator changes, another model or additional fine-tuning is usually required.

ICON aims at a different regime. A single model \(T_\theta\) is trained over a distribution of operators and receives a few in-context examples:

\[\widehat{u}_\tau^{(q)} = T_\theta\!\left(\{(u_0^{(i)},u_\tau^{(i)})\}_{i=1}^{k},\, u_0^{(q)}\right).\]

The hope is that the model infers the operator from these examples and applies it immediately to the query. This lets the model adapt by changing the prompt rather than by changing the weights.

Why do we find this direction compelling?

• Knowledge about a system or task can be supplied at inference time through numerical examples rather than stored entirely in model weights.
• A single frozen model can adapt to many operators and systems simply by changing its context.
• A diverse corpus of numerical systems can improve generalization beyond the systems and even the disciplines seen during training.
• Numerical models can work together with language-model agents: linguistic intelligence interprets goals and organizes inference, while numerical intelligence supplies quantitative predictions.

From a broader perspective, we see three broad stages in how machine learning works with numerical systems. The first stage focused on approximating solution functions, for example Physics-Informed Neural Networks. The second shifted toward approximating solution operators, for example DeepONet and Fourier Neural Operator. ICON points to a third stage: the predictive relation needed for a task is inferred from numerical context and applied by a fixed model. The aim is no longer only to approximate one operator well, but to build models that can acquire and use new numerical knowledge in context.

A Research Thread

Our work on ICON has developed through a sequence of papers, each advancing the framework by one step.

In-Context Operator Learning with Data Prompts for Differential Equation Problems (PNAS 2023)

This paper introduced in-context operator learning and ICON. A single model, without fine-tuning, handled 19 problem types spanning forward and inverse ODE, PDE, and mean-field control settings, with many operators inside each type.

Figure 1: In-context operator learning for a mean-field control problem. The model infers the operator from a few examples and solves the query in one forward pass.

PDE Generalization of In-Context Operator Networks (JCP 2024)

Here we examined whether a single ICON model could generalize across conservation laws with different fluxes and timesteps, including previously unseen PDE forms. We also studied prompt design strategies such as variable transforms and stride manipulation to enlarge the solvable regime.

Fine-Tune Language Models as Multi-Modal Differential Equation Solvers (Neural Networks 2025)

This work adopted a decoder-only, language-model-style architecture and introduced multi-modal prompting. The model can use human language and LaTeX equations alongside numerical examples, providing an early interface between linguistic and numerical forms of context.

Figure 2: Multi-modal in-context operator learning. Textual descriptions and numerical examples can both act as prompt information for the operator.

VICON: Vision In-Context Operator Networks for Multi-Physics Fluid Dynamics Prediction (TMLR 2026)

VICON extends the framework to 2D fields with a patch-wise vision transformer, targeting multi-physics fluid dynamics with flexible rollout and partially missing frames.

GICON: Graph In-Context Operator Networks for Generalizable Spatiotemporal Prediction (arXiv 2026)

GICON brings the same philosophy to graph-structured systems using graph message passing and example-aware positional encoding. It provides a shared representation for numerical observations on irregular domains and studies geometric and example-cardinality generalization on real spatiotemporal problems.

VICX: Generalizable Robot Manipulation via Video Generation and In-Context Operator Network (arXiv 2026, Project)

VICX extends the ICON thread into embodied AI. A frozen video generation model provides high-level visual plans, while V2T-ICON grounds those generated videos into executable robot-state trajectories using retrieved image-state examples as in-context prompts. This turns visual-to-state grounding into an operator inference problem and connects in-context operator learning with closed-loop robot manipulation.

Figure 3: The VICX framework. A frozen video generation model proposes a visual plan, and V2T-ICON grounds it into a robot trajectory using image-state references.

A Foundation Model of Numerical Intelligence with Cross-Disciplinary Generalization (arXiv 2026)

UNICON takes ICON to cross-disciplinary scale and makes the case for numerical intelligence concrete. A single model is trained on numerical systems spanning hydrology, traffic, power systems, weather, land, ocean, soil, solar resources and human mobility, then frozen. Presented with graph-based examples from a new system, it infers the predictive relation expressed by those examples and applies it to new queries. The same model approaches specialist performance even in disciplines absent from training. Combining UNICON with language-model agents to perform contextual ensemble learning (CEL) yields further gains, enabling it to surpass state-of-the-art specialists in a discipline unseen during training. Experiments also show that a more diverse training corpus improves generalization to unseen systems and disciplines.

Figure 4: UNICON learns how to learn from graph-based numerical context, then applies that ability to systems and disciplines not represented in training.

ICON Harnesses: Orchestrating Model Calls at Inference Time

A foundational insight in in-context operator learning is that the same physical problem can be addressed in different ways by reformulating its numerical prompt. Equivalent formulations need not elicit identical behavior from a frozen model: one formulation may place the problem in a regime that the model handles much more reliably than another. This opens a new route to better inference without retraining.

We call the mechanism that exploits this freedom an inference-time harness: an external program around a frozen model that constructs prompts, orchestrates model calls, and processes their outputs without updating the model weights. Across our work, this idea has developed into a continuing research thread:

1. Change of variables and varying stride. In PDE Generalization of In-Context Operator Networks: A Study on 1D Scalar Nonlinear Conservation Laws, change-of-variables and varying-stride strategies reformulate a numerical query before it is given to ICON. They show that prompt-space transformations can extend the range of problems addressed by the same frozen model.

2. Chain of Operators (CHOP). Harness In-Context Operator Learning with Chain of Operators turns prompt reformulation into a compositional harness. CHOP routes numerical prompts through explicit elementary operator transformations, moving a difficult query through intermediate representations where the frozen ICON is more capable and then mapping the result back.

3. Contextual Ensemble Learning (CEL). In A Foundation Model of Numerical Intelligence with Cross-Disciplinary Generalization, CEL orchestrates diverse context-building pathways and fuses their predictions. By constructing contexts from distinct observation slices, CEL systematically exposes the frozen network to complementary facets of the system, fusing these partial views to approximate the target operator far more reliably.

These methods share one principle: after training has produced a capable in-context learner, further adaptation can happen by programming its numerical prompt space. A harness makes the inference behavior of a frozen ICON programmable without updating its weights, separating the costly acquisition of general in-context learning ability from the task-specific orchestration of how that ability is used.

Linguistic Intelligence and Numerical Intelligence

Linguistic and numerical intelligence address different kinds of context. Linguistic intelligence acquires and applies knowledge expressed through words: task descriptions, metadata, scientific concepts, constraints, and domain heuristics. Numerical intelligence acquires and applies knowledge expressed through observations: graph signals, fields, multivariate sequences, and paired examples that reveal how a system behaves.

Their relationship is complementary rather than competitive. A language-model agent can interpret a high-level goal and use semantic or domain knowledge to design an inference harness: it may select examples, construct several contexts, choose transformations, call a frozen ICON repeatedly, and decide how to combine the predictions. The numerical model then learns predictive relations from those contexts and returns quantitative results that language alone cannot reliably produce.

More broadly, this division of labor suggests an AI ecosystem made of interoperable linguistic and numerical building blocks rather than one monolithic model expected to encode every capability.

Figure 5: Linguistic and numerical intelligence as complementary components of an artificial general intelligence ecosystem. From Figure 1a of A Foundation Model of Numerical Intelligence with Cross-Disciplinary Generalization.

A Research Program for Numerical Intelligence

Together, these works define a research program for numerical intelligence: training teaches a model how to learn from context, while context supplies the knowledge required for the problem at hand.

Advancing this program will require shared models, diverse datasets, cross-disciplinary benchmarks, inference methods, and concrete applications. We built icon-core as open-source infrastructure for this community, packaging model implementations, training pipelines, and examples into a reusable stack.