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

Operator learning for partial differential equations (PDEs) on arbitrary geometries builds fast neural surrogates for large-scale simulation. Although recent geometry-adaptive neural operators have made substantial progress, they are mainly designed for forward problems in which inputs and outputs share the same spatial domain. This limits their applicability for boundary value problems (BVPs) and inverse problems, where inputs and outputs may live on different domains. We introduce the Geometry-Adaptive Integral Autoencoder (GAIA), an operator learning model that encodes the domain boundary and the interior field distribution into geometry tokens, and conditions integral transform layers on these tokens via cross-attention, allowing the kernel to adapt locally to geometric features. This yields a single architecture for forward (including BVPs) and inverse problems on arbitrary domains in one pass, without retraining, iterative optimization, or graph construction. We evaluate GAIA on seven 2D and 3D benchmarks, four of which are new or substantially extended benchmarks for inverse problems and BVP: electrical impedance tomography, optical tomography, 3D Darcy flow on varying geometries, and a modified setting of Poisson BVP on mechanical components benchmark (MCB). GAIA sets new state-of-the-art results on every inverse and BVP task, reducing median relative $L^2$ error by 64% on airfoil flow reconstruction and 27% on EIT relative to the next best amortized method, and outperforming all baselines on every shape category of MCB. On other forward problems, GAIA is competitive with specialized solvers while maintaining stable accuracy across point resolutions on which transformer-based baselines degrade.

LookupFFN: Making Transformers Compute-lite for CPU inference

While GPU clusters are the de facto choice for training large deep neural network (DNN) models today, several reasons including ease of workflow, security and cost have led to efforts investigating whether CPUs may be viable for inference in routine use in many sectors of the industry. But the imbalance between the compute capabilities of GPUs and CPUs is huge. Motivated by these considerations, we study a module which is a workhorse within modern DNN architectures, GEMM based Feed Forward Networks (FFNs), and assess the extent to which it can be made compute- (or FLOP-) lite. Specifically, we propose an alternative formulation (we call it LookupFFN) to GEMM based FFNs inspired by the recent studies of using Locality Sensitive Hashing (LSH) to approximate FFNs. Our formulation recasts most essential operations as a memory look-up, leveraging the trade-off between the two resources on any platform: compute and memory (since CPUs offer it in abundance). For RoBERTa language model pretraining, our formulation achieves similar performance compared to GEMM based FFNs, while dramatically reducing the required FLOP. Our development is complemented with a detailed hardware profiling of strategies that will maximize efficiency -- not just on contemporary hardware but on products that will be offered in the near/medium term future. Code is avaiable at \url{https://github.com/mlpen/LookupFFN}.