Pith Number

pith:O3A2XU56

pith:2026:O3A2XU56KWBBBREARYUJ4GFPJL

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A deep backward regression-based scheme for high-dimensional nonlinear partial differential equations

Qiang Han, Shaolin Ji, Yunzhang Li

The deep backward regression scheme solves high-dimensional nonlinear PDEs by turning stochastic losses into deterministic conditional expectations.

arxiv:2603.14721 v2 · 2026-03-16 · math.NA · cs.NA

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4 Citations open

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Claims

C1strongest claim

Numerical experiments demonstrate that the DBR scheme consistently outperforms the DBDP1 method; notably, for complex unbounded PDEs, DBR maintains high accuracy in regimes where DBDP1 fails to converge beyond d=10. Theoretically, we derive rigorous upper error bounds and establish half-order convergence for the proposed scheme.

C2weakest assumption

The transformation of simulated backward stochastic difference equations into their conditional expectation representations can be accurately approximated by neural networks without introducing bias that invalidates the error bounds or the observed stability gains.

C3one line summary

A new DBR algorithm reformulates backward stochastic difference equations via conditional expectations to reduce variance and improve accuracy for high-dimensional nonlinear parabolic PDEs, outperforming DBDP1 beyond dimension 10.

References

38 extracted · 38 resolved · 0 Pith anchors

[1] Anil, C., Lucas, J. and Grosse, R. (2019).Sorting out Lipschitz function approximation. In International conference on machine learning (pp. 291-301). PMLR 2019

[2] Bouchard, B. and Touzi, N. (2004).Discrete-time approximation and Monte-Carlo simulation of back- ward stochastic differential equations. Stoch. Process. Their Appl., 111(2), 175-206 2004

[3] Bouchard, B. and Chassagneux, J. F. (2008).Discrete-time approximation for continuously and dis- cretely reflected BSDEs.Stoch. Process. Their Appl, 118(12), 2269-2293 2008

[4] Bungartz, H. J. and Griebel, M. (2004).Sparse grids. Acta Numer., 13, 147-269. 29 2004

[5] Cai, W., Fang, S. and Zhou, T. (2025).SOC-MartNet: A martingale neural network for the Hamilton- Jacobi-Bellman equation without explicit in stochastic optimal controls. SIAM J. Sci. Comput., 47(4), C 2025

Formal links

1 machine-checked theorem link

Receipt and verification

First computed	2026-05-18T03:09:22.698942Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

76c1abd3be558210c4808e289e18af4aec50b8cba15a0143f0b22b60d2270f3a

Aliases

arxiv: 2603.14721 · arxiv_version: 2603.14721v2 · doi: 10.48550/arxiv.2603.14721 · pith_short_12: O3A2XU56KWBB · pith_short_16: O3A2XU56KWBBBREA · pith_short_8: O3A2XU56

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Verify this Pith Number yourself

curl -sH 'Accept: application/ld+json' https://pith.science/pith/O3A2XU56KWBBBREARYUJ4GFPJL \
  | 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: 76c1abd3be558210c4808e289e18af4aec50b8cba15a0143f0b22b60d2270f3a

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "7f5cd1a801b90546c57714e9ecd1cca8c5be324666c1e4e26faa8acafa4f68e5",
    "cross_cats_sorted": [
      "cs.NA"
    ],
    "license": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
    "primary_cat": "math.NA",
    "submitted_at": "2026-03-16T01:54:09Z",
    "title_canon_sha256": "85b882af94b8e101cf1ab7e52692e60f7d8db79f5a52ce60058fd3a8501ba011"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2603.14721",
    "kind": "arxiv",
    "version": 2
  }
}