RT @BaigYasa: Was extremely fun to work on this paper with @jerrywliu and finally fulfilling our 7 year plan from year one of undergrad to….

@BaigYasa @rajat_vd @HazyResearch 11/10.BWLer was just presented at the Theory of AI for Scientific Computing (TASC) workshop at COLT 2025, where it received Best Paper 🏆. Huge thanks to the organizers (@nmboffi, @khodakmoments, Jianfeng Lu, @__tm__157, @risteski_a) for a fantastic event!

10/10.BWLer is just the beginning – we're excited to build precise, generalizable ML models for PDEs & physics!.📄 Paper: 🧠 Blog: 💻 Code: w/ @BaigYasa, Denise Lee, @rajat_vd, Atri Rudra, @HazyResearch.

9/10.Of course, there’s no free lunch. Like spectral methods, BWLer struggles with discontinuities or irregular domains – sometimes taking hours to match the RMSE of PINNs that train in minutes. We view BWLer as a proof-of-concept toward high-precision scientific ML! 🔬

8/10.Explicit BWLer can go even further 🚀.With a second-order optimizer, it reaches 10⁻¹² RMSE – near float64 machine precision! – and up to 10 billion× lower error than standard MLPs. See comparison across benchmark PDEs ⬇️

7/10.Adding BWLer-hats 🎩 to standard MLPs improves RMSE by up to 1800× across benchmark PDEs (convection, reaction, wave). Why? BWLer’s global derivatives encourage smoother, more coherent solutions. Example below: standard MLP vs BWLer-hatted on the convection equation ⬇️

6/10.BWLer comes in two modes:.– BWLer-hat 🎩: adds an interpolation layer atop an NN.– Explicit BWLer 🎳: replaces the NN, learns function values directly.Both versions let us explicitly tune expressivity and conditioning – yielding big precision gains on benchmark PDEs 📈

5/10.💡If polynomials work so well, why not use them for PINNs?.We introduce BWLer 🎳, a drop-in module for physics-informed learning. Built on barycentric polynomials, BWLer wraps or replaces MLPs with a numerically stable interpolant rooted in classical spectral methods.

4/10.Turns out, MLPs struggle 😬– even on simple sinusoids. Despite 1000× more parameters, they plateau far above machine precision, with RMSE up to 10,000× worse than basic polynomial interpolants!

3/10.We strip away the PDE constraints and ask a simpler question: how well can MLPs perform basic interpolation? 🧩.E.g. can MLPs recover the black curve just from the red training points? (Pictured: f(x) = sin(4x).)

2/10.Physics-Informed Neural Networks (PINNs) solve PDEs by training a neural network to satisfy the equation – no mesh required and handles complex boundaries/geometries with ease. But despite their flexibility, PINNs often fall short on precision, even on simple 2D problems.

1/10.ML can solve PDEs – but precision🔬is still a challenge. Towards high-precision methods for scientific problems, we introduce BWLer 🎳, a new architecture for physics-informed learning achieving (near-)machine-precision (up to 10⁻¹² RMSE) on benchmark PDEs. 🧵How it works:

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