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GAIA: Geometry-Adaptive Operator Learning for Forward and Inverse Problems

NeurIPS 2026 · Probabilistic and scientific modeling › Differential equations and PDEs · approach: Neural operators for general geometries · 0 citations (OpenAlex)

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Proposes GAIA, a geometry-adaptive integral autoencoder that conditions integral kernels on boundary and interior-field tokens via cross-attention, enabling single-pass forward, BVP and inverse operator learning on arbitrary domains.

Problem

Neural operators for PDEs on varying geometries mostly handle forward problems with shared input/output domains; boundary value and inverse problems with differing input and output domains remain poorly served without iterative optimization.

Approach

Encode domain boundary (PointNet-style) and interior field (soft-cluster slice tokens) into geometry tokens, and condition IAE-Net integral kernels on these tokens via cross-attention so the kernel adapts locally to geometry; decoupled encoder/decoder query sets let inputs and outputs live on different domains.

Key results

• On inverse tasks GAIA reaches 0.71% median relative L2 on EIT (vs 0.97% LNO) and 0.58% on Airfoil (vs 1.60% CORAL), a 64% reduction on Airfoil.
• On the MCB Poisson BVP GAIA gets best error on all four shape categories, e.g., 3.44% on gears vs 5.89% for NGF.
• On 3D Darcy flow Transolver (0.73%) beats GAIA (1.11%) at the training resolution, while GAIA maintains accuracy under resolution changes on Elasticity.

Surprising findings

• Transolver and GAOT degrade by an order of magnitude at coarser evaluation resolutions on Elasticity, where they look strongest in-distribution.
• Cross-attention conditioning gives an order-of-magnitude gain over concatenation on Airfoil (0.58% vs 5.91%).

Limitations

• Designed for time-independent operators; time-dependent extension left to future work.
• Trades some in-distribution accuracy on shared-mesh forward problems (e.g., 3D Darcy) for discretization invariance.
• Cross-attention cost scales as O(NK), which may need latent compression for very large point clouds (N > 1e6).

Experimental setup

Datasets: Poisson-Gauss Elasticity 3D Darcy flow MCB (mechanical components benchmark) EIT Optical tomography Airfoil (NACA transonic)

Benchmarks: EIT Optical tomography Airfoil MCB Poisson BVP Poisson-Gauss Elasticity 3D Darcy flow

Metrics: median relative L2 error latency throughput peak memory

Baselines: GINO CORAL Transolver GAOT LNO NGF NIO

Builds on: IAE-Net Transolver PointNet Fourier features

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