# BWLer: Barycentric Weight Layer Elucidates a Precision-Conditioning Tradeoff for PINNs

arXiv: https://arxiv.org/abs/2506.23024 · HTML: https://mlsanity.com/paper/2506.23024/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 13 of its 37 references that are in the map.
- First posted on arXiv 28 Jun 2025.

## Builds on (most cited first, 13 of 13)

- [Convex optimization](https://mlsanity.com/paper/2106.01946/) (2021)
- [Characterizing possible failure modes in physics-informed neural networks](https://mlsanity.com/paper/2109.01050/) (2021)
- [PyHessian: Neural Networks Through the Lens of the Hessian](https://mlsanity.com/paper/1912.07145/) (2019)
- [Machine Learning of Linear Differential Equations using Gaussian Processes](https://mlsanity.com/paper/1701.02440/) (2017)
- [Estimates on the generalization error of Physics Informed Neural Networks (PINNs) for approximating PDEs](https://mlsanity.com/paper/2006.16144/) (2020)
- [Weak baselines and reporting biases lead to overoptimism in machine learning for fluid-related partial differential equations](https://mlsanity.com/paper/2407.07218/) (2024)
- [PINNacle: A Comprehensive Benchmark of Physics-Informed Neural Networks for Solving PDEs](https://mlsanity.com/paper/2306.08827/) (2023)
- [Multi-stage Neural Networks: Function Approximator of Machine Precision](https://mlsanity.com/paper/2307.08934/) (2023)
- [Implicit Neural Spatial Representations for Time-dependent PDEs](https://mlsanity.com/paper/2210.00124/) (2022)
- [Data-Driven Aerospace Engineering: Reframing the Industry with Machine Learning](https://mlsanity.com/paper/2008.10740/) (2020)
- [Randomized Sparse Neural Galerkin Schemes for Solving Evolution Equations with Deep Networks](https://mlsanity.com/paper/2310.04867/) (2023)
- [Precision Machine Learning](https://mlsanity.com/paper/2210.13447/) (2022)
- [Towards Learning High-Precision Least Squares Algorithms with Sequence Models](https://mlsanity.com/paper/2503.12295/) (2025)

## Built on by (most cited first, 1 of 1)

- [PINNs Failure Modes are Overfitting](https://mlsanity.com/paper/2605.30910/) (2026)

## Abstract (arXiv)

Physics-informed neural networks (PINNs) offer a flexible way to solve partial differential equations (PDEs) with machine learning, yet they still fall well short of the machine-precision accuracy many scientific tasks demand. In this work, we investigate whether the precision ceiling comes from the ill-conditioning of the PDEs or from the typical multi-layer perceptron (MLP) architecture. We introduce the Barycentric Weight Layer (BWLer), which models the PDE solution through barycentric polynomial interpolation. A BWLer can be added on top of an existing MLP (a BWLer-hat) or replace it completely (explicit BWLer), cleanly separating how we represent the solution from how we take derivatives for the PDE loss. Using BWLer, we identify fundamental precision limitations within the MLP: on a simple 1-D interpolation task, even MLPs with O(1e5) parameters stall around 1e-8 RMSE -- about eight orders above float64 machine precision -- before any PDE terms are added. In PDE learning, adding a BWLer lifts this ceiling and exposes a tradeoff between achievable accuracy and the conditioning of the PDE loss. For linear PDEs we fully characterize this tradeoff with an explicit error decomposition and navigate it during training with spectral derivatives and preconditioning. Across five benchmark PDEs, adding a BWLer on top of an MLP improves RMSE by up to 30x for convection, 10x for reaction, and 1800x for wave equations while remaining compatible with first-order optimizers. Replacing the MLP entirely lets an explicit BWLer reach near-machine-precision on convection, reaction, and wave problems (up to 10 billion times better than prior results) and match the performance of standard PINNs on stiff Burgers' and irregular-geometry Poisson problems. Together, these findings point to a practical path for combining the flexibility of PINNs with the precision of classical spectral solvers.