Pith Number

pith:WVPUXISA

pith:2026:WVPUXISAULDBP7GD6YEILOSMGN

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Breakeven complexity: A new perspective on neural partial differential equation solvers

Mikhail Khodak, Nicholas Roberts, Tanya Marwah, Yijing Zhang

Neural PDE solvers reach cost parity with traditional methods after fewer solves as problems grow harder, higher-dimensional, or higher-Reynolds.

arxiv:2605.15399 v1 · 2026-05-14 · cs.LG · cs.AI · cs.NA · math.NA · physics.comp-ph

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Record completeness

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3 Author claim open · sign in to claim

4 Citations open

5 Replications open

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The bundle contains the canonical record plus signed events. A mirror can host it anywhere and recompute the same current state with the deterministic merge algorithm.

Claims

C1strongest claim

our results suggest that neural PDE solvers become more effective as problems get harder in terms of cost, dimension, rollout, physics regime (e.g. higher Reynolds number), etc.

C2weakest assumption

That scaling laws can be reliably applied to allocate training budget between data generation and model training while achieving smooth error-matching between neural and traditional solvers across diverse PDE settings and regimes.

C3one line summary

Breakeven complexity is introduced to evaluate neural PDE solvers by total end-to-end cost, with results indicating they become advantageous for harder problems such as higher dimensions, longer rollouts, and higher Reynolds numbers.

References

38 extracted · 38 resolved · 1 Pith anchors

[1] Abnar, S., H. Shah, D. Busbridge, A. El-Nouby, J. M. Susskind, and V . Thilak (2025). Parameters vs FLOPs: Scaling laws for optimal sparsity for mixture-of-experts language models. InForty- second Int 2025

[2] Ahrens, J. P., B. Geveci, and C. C. Law (2005). ParaView: An end-user tool for large-data visualization. InThe Visualization Handbook 2005

[3] Fluid intelligence: A forward look on AI foundation models in computational fluid dynamics 2025

[4] Bochev, P. B. and D. Ridzal (2009). An optimization-based approach for the design of pde solution algorithms.SIAM journal on numerical analysis 47(5), 3938–3955 2009

[5] Choudhary, N., V . Singh, A. Talwalkar, N. M. Boffi, M. Khodak, and T. Marwah (2025). Pre-generating multi-difficulty pde data for few-shot neural pde solvers.arXiv preprint arXiv:2512.00564 2025

Formal links

2 machine-checked theorem links

Receipt and verification

First computed	2026-05-20T00:00:56.664930Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

b55f4ba240a2c617fcc3f60885ba4c3345393579bcdcb24bbd2e0085cd2aa190

Aliases

arxiv: 2605.15399 · arxiv_version: 2605.15399v1 · doi: 10.48550/arxiv.2605.15399 · pith_short_12: WVPUXISAULDB · pith_short_16: WVPUXISAULDBP7GD · pith_short_8: WVPUXISA

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Verify this Pith Number yourself

curl -sH 'Accept: application/ld+json' https://pith.science/pith/WVPUXISAULDBP7GD6YEILOSMGN \
  | jq -c '.canonical_record' \
  | python3 -c "import sys,json,hashlib; b=json.dumps(json.loads(sys.stdin.read()), sort_keys=True, separators=(',',':'), ensure_ascii=False).encode(); print(hashlib.sha256(b).hexdigest())"
# expect: b55f4ba240a2c617fcc3f60885ba4c3345393579bcdcb24bbd2e0085cd2aa190

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "77471c699f1120652257fe9c4d64b281cb4922b846f4a061c7daa8ceda8bb13e",
    "cross_cats_sorted": [
      "cs.AI",
      "cs.NA",
      "math.NA",
      "physics.comp-ph"
    ],
    "license": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
    "primary_cat": "cs.LG",
    "submitted_at": "2026-05-14T20:33:24Z",
    "title_canon_sha256": "8e662b00ad6bc5c4121670df8f2ccac7cd1bc07754f22446887f74f502886c24"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2605.15399",
    "kind": "arxiv",
    "version": 1
  }
}