Multitask Hamiltonian Learning across Levels of Theory

Multitask Hamiltonian Learning across Levels of Theory

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$; the standard number fields have their usual meanings. All additional conventions are stated locally.

A learned Hamiltonian is an effective operator whose parameters are fitted to electronic-structure data. Distinct levels of theory supply tasks with different approximations and errors. Multitask learning shares selected parameters or representations while allowing task-dependent outputs. Transfer benefit is assessed against corresponding single-task prediction and sample requirements.

Catalogue wording retained for traceability.

Develop and analyze a multitask learning framework for Hamiltonians trained on data generated at different levels of electronic-structure theory. Determine how information should be shared across tasks and when transfer improves accuracy and sample efficiency.

Scope and source qualification.

The source gives a learning-framework question, not a universal theorem that sharing always improves every task.

Short English statement

Develop Hamiltonian learning that shares information across electronic-structure approximations and determine when that sharing improves prediction and sample efficiency.

Research attempt

Model, date and investigated scope.

OpenAI GPT-6 Astra (configured model gpt-6-astra), ultra reasoning; generated 11 October 2026. Computed exact transfer risk and a necessary/sufficient pooling improvement condition in a known-basis scalar Hamiltonian model.

Sources and literature check.

Read the full supplied definition, catalogue wording, source qualifications and listed references for 30006232. No fresh status or novelty search was performed. The argument below proves its auxiliary statements directly and does not import an unverified external theorem.

A one-parameter Hamiltonian transfer calculation.

Fix a known Hermitian matrix $A$ with Frobenius norm one and let task $j$ have Hamiltonian $H_j=\theta_j A$. Suppose $n$ independent scalar observations per task equal $\theta_j$ plus centered noise of variance $\sigma^2$, independently between tasks. Their sample means $\bar Y_j$ have variance $v=\sigma^2/n$. Estimate task one by $\widehat\theta_1=(1-\lambda)\bar Y_1+\lambda\bar Y_2$, $0\leq\lambda\leq1$. With $\Delta=\theta_2-\theta_1$, direct bias/variance decomposition gives \[ \mathbb E\|\widehat\theta_1A-H_1\|_F^2 =v(1-\lambda)^2+v\lambda^2+\lambda^2\Delta^2. \] Equal pooling, $\lambda=1/2$, improves on task-one-only risk $v$ exactly when $\Delta^2<2v$.

Oracle sharing and negative transfer.

If $v=\Delta=0$, every $\lambda$ has zero risk. Otherwise differentiation of the quadratic risk yields $\lambda_*=v/(2v+\Delta^2)$. This is an oracle choice because $\Delta$ is unknown. If $\Delta=0$, pooling halves the risk. If the task discrepancy is fixed and $n\to\infty$, equal pooling retains bias $\Delta^2/4$ while the independent estimator becomes accurate, so unconditional sharing eventually harms the target task.

Full learning gap.

The example preserves Hermiticity but assumes direct coefficient observations in a known common basis. Electronic-structure data are usually more indirect and task discrepancy can vary with configuration and operator component. The first missing step is identifying estimable shared structure with a data-dependent transfer rule and error control under those observation maps. No empirical comparison or general multitask Hamiltonian-learning theorem is supplied; the calculation only establishes a transparent sharing-versus-bias diagnostic.

Verification and adversarial audit.

Checked Frobenius normalization, independent-noise variance, pooling threshold, oracle optimum and the large-sample negative-transfer limit. Model self-audit only; no independent human mathematical verification. Applied contribution prompt A before drafting and B afterwards. Checked Frobenius normalization, independent-noise variance, pooling threshold, oracle optimum and the large-sample negative-transfer limit. First remaining gap: Identify and estimate shared operator structure from source electronic-structure observations with a data-dependent transfer guarantee and controlled task bias. The inherited source status is preserved; original-target completion remains zero. Additional GPT-6 Astra reviewer (ultra reasoning): Computed the estimator bias lambda Delta and variance v[(1-lambda)²+lambda²] using independent tasks and n independent observations per task. Checked equal-pooling risk v/2+Delta²/4 and the strict improvement condition Delta²<2v. Differentiated risk to obtain lambda*=v/(2v+Delta²) when the denominator is positive; it lies in [0,1/2]. Checked the fixed nonzero task-discrepancy bias Delta²/4 as n tends to infinity and the declared restriction to direct known-basis coefficient observations. The requested zero-variance, zero-discrepancy boundary case was added by the author and its presence checked by the coordinator.

Tools, process and contributions.

Python 3 and local file tools used to read definitions and serialize manuscripts. Calculations are symbolic reasoning; no numerical experiment or theorem prover. Contribution prompt A was followed by prompt B. GPT-6 Astra with ultra reasoning drafted the research and performed its stated self-audit; any additional coordinator contribution is identified in the verification and audit. A GPT-6-based Codex coordinator assembled the manuscript and checked repository integration; its exact variant and reasoning configuration were not exposed. No human mathematical contribution, independent human proof check or formal proof certification is supplied. Complete derivations and the consulted source references are included here; private reasoning transcripts are not published.

Final status and first remaining gap.

UNRESOLVED. Identify and estimate shared operator structure from source electronic-structure observations with a data-dependent transfer guarantee and controlled task bias.

Completion Estimate

COMPLETION: 0% This is an elementary auxiliary calculation or an explicit test of a restricted strategy. No new portion of the full source target is established.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath, record 30006232 (OWR-14299094-002). Catalogue curation under CC BY 4.0; definitions and English exposition added locally. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Mathematical Methods in Quantum Chemistry $2025$, Oberwolfach Reports, DOI 10.4171/owr/2025/15, pp. 615--688. Original report and mathematical context. Incorporated report material remains under the source CC BY-SA 4.0 terms unless otherwise noted. https://doi.org/10.4171/owr/2025/15.