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Transformers for In-Context PDE Solving and Inverse Problems transformers for inverse problems on PDEs, Part 6: How Attention Learns a Preconditioner
Which training curves are exact, which are local diagnostics, and how query-key-value matrices become a learned solver step.
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Muon for Phase Retrieval IV: Proof Map for Signal Recovery
Which steps are algebraic, which come from high-dimensional risk-curve theory, and where the final random-matrix comparison remains.
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Transformers for In-Context PDE Solving and Inverse Problems transformers for inverse problems on PDEs, Part 5: Separating Encoder, Decoder, and Generalization Error
A readable error budget for task inference, solver depth, training tasks, and held-out generalization.
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Muon for Phase Retrieval III: Choosing the Muon Power During Training
Why the online exponent should move with the spectral front, and how entropy-regularized control turns spectral margins into a trajectory.
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Transformers for In-Context PDE Solving and Inverse Problems transformers for inverse problems on PDEs, Part 4: What Can Be Proved About the Solver?
The finite-dimensional encoder and decoder certificates, and why a full training theory needs replica order parameters.
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Transformers for In-Context PDE Solving and Inverse Problems transformers for inverse problems on PDEs, Part 3: Turning PDE Solutions Into Transformer Tokens
A plain-language pipeline: functions become vectors, prompts become weak equations, and decoder layers become solver steps.
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Dynamic BBP for Multi-Index Phase Retrieval
A self-contained note on when the empirical Hessian reveals teacher directions in quadratic multi-index phase retrieval.
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Transformers for In-Context PDE Solving and Inverse Problems transformers for inverse problems on PDEs, Part 2: Inferring Unknown PDE Coefficients From Examples
When the PDE operator changes across tasks, the prompt becomes an inverse problem for the hidden coefficient vector.
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Muon for Phase Retrieval I: Recovering a Hidden Signal From Quadratic Measurements
Why fixed-power Muon turns phase retrieval into a moving spectral-front problem.
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Transformers for In-Context PDE Solving and Inverse Problems transformers for inverse problems on PDEs, Part 1: Solving Many PDE Inputs With One Learned Decoder
Start here: the PDE operator is fixed, so the transformer decoder is tested as a reusable preconditioned solver.
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A simple proof of an ENS Ulm exercise using Liouville's theorem
A simple proof using discrete harmonic functions and Liouville's theorem
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A simplified proof of an integral identity using density arguments
A simplified proof of Theorem 2 using density arguments, avoiding lengthy variable changes