PDE-Constrained Optimization (PDECO) problems can be accelerated significantly by employing gradient-based methods with surrogate models like neural operators compared to traditional numerical solvers. However, this approach faces two key challenges:(1) Data inefficiency: Lack of efficient data sampling and effective training for neural operators, particularly for optimization purpose.(2) Instability: High risk of optimization derailment due to inaccurate neural operator predictions and gradients.To address these challenges, we propose a novel framework: (1) Optimization-oriented training: we leverage data from full steps of traditional optimization algorithms and employ a specialized training method for neural operators. (2) Enhanced derivative learning: We introduce a Virtual-Fourier layer to enhance derivative learning within the neural operator, a crucial aspect for gradient-based optimization. (3) Hybrid optimization: We implement a hybrid approach that integrates neural operators with numerical solvers, providing robust regularization for the optimization process.Our extensive experimental results demonstrate the effectiveness of our model in accurately learning operators and their derivatives. Furthermore, our hybrid optimization approach exhibits robust convergence.

Imagine you're trying to optimize something complex – like making a car part as light as possible while still being strong, or designing a chemical process to maximize output. Often, the behavior of these systems is governed by complex physics described by Partial Differential Equations (PDEs). When we try to find the best design or process under these rules, we're doing what's called PDE-Constrained Optimization (PDECO). Solving these optimization problems using traditional computer simulations is incredibly slow and computationally expensive.---### The Need for Speed (and Trust)Our research aims to accelerate these PDECO problems using AI models called "neural operators." These AI tools can learn system behaviors much faster. However, two main challenges arise:1. Data Inefficiency: AI models usually need vast amounts of data. How do we train them effectively without endless slow simulations?2. Instability: AI predictions can be inaccurate, risking faulty designs or optimization failures. We need reliability.---### Our Solution: Smart, Reliable AIWe developed a new framework to address this:1. Smarter Training: We train our AI by showing it how reliable, traditional PDECO methods work step-by-step, making it learn efficiently from less data.2. Accurate Calculations: For optimization, precise calculations (like "derivatives" which show how things change) are crucial. We added a special "Virtual-Fourier" layer to our AI to ensure it learns these accurately.3. Hybrid Approach: We combine our fast AI with robust traditional solvers. The AI speeds things up, but if it makes a questionable prediction, the reliable traditional solver steps in to maintain accuracy and prevent errors. It's like having a fast AI assistant checked by an expert.---### The OutcomeOur results show our model accurately learns complex behaviors and their derivatives. Crucially, our hybrid approach ensures robust and reliable convergence, meaning faster, more confident solutions for these challenging PDECO problems.