Deep vs. Shallow: Benchmarking Physics-Informed Neural Architectures on the Biharmonic Equation

Partial differential equation (PDE) solvers are fundamental to engineering simulation. Classical mesh-based approaches (finite difference/volume/element) are fast and accurate on high-quality meshes but struggle with higher-order operators and complex, hard-to-mesh geometries. Recently developed physics-informed neural networks (PINNs) and their variants are mesh-free and flexible, yet compute-intensive and often less accurate. This paper systematically benchmarks RBF-PIELM, a rapid PINN variant-an extreme learning machine with radial-basis activations-for higher-order PDEs. RBF-PIELM replaces PINNs' time-consuming gradient descent with a single-shot least-squares solve. We test RBF-PIELM on the fourth-order biharmonic equation using two benchmarks: lid-driven cavity flow (streamfunction formulation) and a manufactured oscillatory solution. Our results show up to $(350\times)$ faster training than PINNs and over $(10\times)$ fewer parameters for comparable solution accuracy. Despite surpassing PINNs, RBF-PIELM still lags mature mesh-based solvers and its accuracy degrades on highly oscillatory solutions, highlighting remaining challenges for practical deployment.

Towards Fast Option Pricing PDE Solvers Powered by PIELM

Partial differential equation (PDE) solvers underpin modern quantitative finance, governing option pricing and risk evaluation. Physics-Informed Neural Networks (PINNs) have emerged as a promising approach for solving the forward and inverse problems of partial differential equations (PDEs) using deep learning. However they remain computationally expensive due to their iterative gradient descent based optimization and scale poorly with increasing model size. This paper introduces Physics-Informed Extreme Learning Machines (PIELMs) as fast alternative to PINNs for solving both forward and inverse problems in financial PDEs. PIELMs replace iterative optimization with a single least-squares solve, enabling deterministic and efficient training. We benchmark PIELM on the Black-Scholes and Heston-Hull-White models for forward pricing and demonstrate its capability in inverse model calibration to recover volatility and interest rate parameters from noisy data. From experiments we observe that PIELM achieve accuracy comparable to PINNs while being up to $30\times$ faster, highlighting their potential for real-time financial modeling.

RE-GAINS & EnCHANT: Intelligent Tool Manipulation Systems For Enhanced Query Responses

Despite the remarkable success of LLMs, they still suffer from tool invocation and tool chaining due to inadequate input queries and/or tool argument descriptions. We propose two novel frameworks, RE-GAINS and EnCHANT, enabling LLMs to tackle tool manipulation for solving complex user queries by making API calls. EnCHANT is an open-source solution that makes use of an LLM format enforcer, an LLM(OpenChat 3.5) and a retriever(ToolBench's API Retriever). RE-GAINS is based on OpenAI models and embeddings using a special prompt based on the RAP paper. Both solutions cost less than $0.01 per query with minimal latency, therefore showcasing the usefulness of the frameworks.