Coefficient-to-Basis Network: A Fine-Tunable Operator Learning Framework for Inverse Problems with Adaptive Discretizations and Theoretical Guarantees

TL;DR

C2BNet introduces a two-component operator-learning framework for inverse PDE problems, combining a coefficient network and a basis network to learn mappings from PDE solutions to inverse QoIs. The method enables seamless adaptation to new discretizations via fine-tuning primarily the final basis layer, significantly reducing retraining cost while preserving accuracy. Theoretical results establish approximation and generalization bounds that depend on the intrinsic dimensionality of inputs and the targeted output subspace, without requiring explicit encoders for all data. Numerical experiments on RTE, elliptic, and time-dependent diffusion problems demonstrate fast, robust performance and effective transfer to finer discretizations, underscoring C2BNet’s practical utility in scientific computing and engineering applications.

Abstract

We propose a Coefficient-to-Basis Network (C2BNet), a novel framework for solving inverse problems within the operator learning paradigm. C2BNet efficiently adapts to different discretizations through fine-tuning, using a pre-trained model to significantly reduce computational cost while maintaining high accuracy. Unlike traditional approaches that require retraining from scratch for new discretizations, our method enables seamless adaptation without sacrificing predictive performance. Furthermore, we establish theoretical approximation and generalization error bounds for C2BNet by exploiting low-dimensional structures in the underlying datasets. Our analysis demonstrates that C2BNet adapts to low-dimensional structures without relying on explicit encoding mechanisms, highlighting its robustness and efficiency. To validate our theoretical findings, we conducted extensive numerical experiments that showcase the superior performance of C2BNet on several inverse problems. The results confirm that C2BNet effectively balances computational efficiency and accuracy, making it a promising tool to solve inverse problems in scientific computing and engineering applications.

Figures (11)

• Figure 1: (a) An illustration of C2BNet in (\ref{['eq.psi.archi']}). (b) Fine-tuning from discretization $S_{\mathcal{Y}}$ to $S'_{\mathcal{Y}}$.
• Figure 2: Demonstration of two scatter parameters for RTE. Each scattering field has 5 channels and each channels has two degrees of freedom width and height.
• Figure 3: RTE results. Left: Relative error decay with respect to the number of training samples. Right: Error decay in log-scale. Note a linear line is fit to better demonstrate the linear decay in log.
• Figure 4: Demonstration of two permeability realizations for elliptic equation. Each permeability is determined by 4 degrees of freedom.
• Figure 5: Elliptic example. Left: Error decay with respect to the number of training samples. Right: Error decay in logarithmic scale. We infer the inverse QoIs, which correspond to the permeability discretized on a $10\times 10$ mesh. To better illustrate the convergence rate, we fit a linear line to the error in the logarithmic scale on the right.

Theorems & Definitions (11)

• Definition 1: Chart
• Definition 2: $C^k$ Atlas
• Definition 3: $C^s$ functions on $\mathcal{M}$
• Definition 4: Reach federer1959curvatureniyogi2008finding
• Theorem 1
• Corollary 1
• Theorem 2
• Lemma 1
• Lemma 2
• Lemma 3: Lemma 11 of liu2025generalization