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Generalization Limits of In-Context Operator Networks for Higher-Order Partial Differential Equations

Jamie Mahowald, Tan Bui-Thanh

Abstract

We investigate the generalization capabilities of In-Context Operator Networks (ICONs), a new class of operator networks that build on the principles of in-context learning, for higher-order partial differential equations. We extend previous work by expanding the type and scope of differential equations handled by the foundation model. We demonstrate that while processing complex inputs requires some new computational methods, the underlying machine learning techniques are largely consistent with simpler cases. Our implementation shows that although point-wise accuracy degrades for higher-order problems like the heat equation, the model retains qualitative accuracy in capturing solution dynamics and overall behavior. This demonstrates the model's ability to extrapolate fundamental solution characteristics to problems outside its training regime.

Generalization Limits of In-Context Operator Networks for Higher-Order Partial Differential Equations

Abstract

We investigate the generalization capabilities of In-Context Operator Networks (ICONs), a new class of operator networks that build on the principles of in-context learning, for higher-order partial differential equations. We extend previous work by expanding the type and scope of differential equations handled by the foundation model. We demonstrate that while processing complex inputs requires some new computational methods, the underlying machine learning techniques are largely consistent with simpler cases. Our implementation shows that although point-wise accuracy degrades for higher-order problems like the heat equation, the model retains qualitative accuracy in capturing solution dynamics and overall behavior. This demonstrates the model's ability to extrapolate fundamental solution characteristics to problems outside its training regime.
Paper Structure (24 sections, 14 equations, 9 figures)

This paper contains 24 sections, 14 equations, 9 figures.

Table of Contents

  1. Introduction
  2. In-context operator learning for differential equation
  3. In-context operator networks
  4. In-context operator learning with data prompts for differential equation problems
  5. Language model integration for multi-modal differential equation solving
  6. Conservation law applications in PDE contexts
  7. Problem setup
  8. Neural vs. differential operators
  9. Data generation and model architecture
  10. Synthetic data generation using numerical methods
  11. Data-generation using traditional numerical solvers
  12. Data-generation by starting with the solution $u$
  13. Model architecture
  14. Results
  15. Results and analysis
  16. ...and 9 more sections

Figures (9)

  • Figure 1: Two samples from a single operator defined by $(a, ..., f)$. In this case $(a,b,c,d,e,f) = (0.4563, 0.1500, -0.4341, -0.0525, -0.0457, 0.1578)$. These were generated from a Gaussian process with an RBF kernel with length scale 0.2 (one fifth of the domain) and variance 2.0.
  • Figure 2: We achieve similar average error as the original paper on inference across context sizes. Left: average across all 19 problems of the average testing error for each problem (i.e., the average of the averages), with similarly defined error bars. This can be compared to the original experiment's error icon_1_original on the right.
  • Figure 3: Average error in inference vs. the number of samples provided, split into three panels for visibility. See \ref{['appendix:ode-pde-forms']} for full forms. We see that forward problems tend to achieve higher accuracy than their inverse counterparts. In the ICON setting, accuracy is uncorrelated with the complexity of the relationship: intricate MFC problems achieve similar or lower error as simpler ODEs, while intermediate PDEs have a wide range.
  • Figure 4: The model is kept from making entirely random predictions by the transformer's accurate detection of global patterns, sometimes at the expense of local shapes. The model input is shown on the left, where individual input points are randomly selected (black squares). The model is expected to infer on future points based on this context. While the model fails to discern individual features (third panel), its average predictions follow those of the ground truths they aim to predict. The mean-squared difference between the averages of the prediction and the ground truth (bottom right) is much lower than the error in individual tokens (bottom left), computed as a Frobenius norm of the element-wise difference between the prediction and ground-truth matrices.
  • Figure 5: Inference on several problems: (a) forward damped-oscillator, where the model is supplied with example conditions and quantities of interest (light colors) and the question condition (bright red) and is expected to predict the question quantity of interest (dark blue), with ground-truth QoI shown in black; (b) forward mean-field control problem; (c) forward ordinary differential equation; (d) backward ordinary differential equation.
  • ...and 4 more figures