Efficient Deep Learning Methods for Solving High-dimensional Partial Differential Equations for Applications in Option Pricing
Bibliographic record
Abstract
Partial Differential Equations (PDEs) are used to model a variety of dynamical systems around us. Recent advances in deep learning have enabled us to solve these PDEs in higher dimensions by addressing the Curse of Dimensionality (COD). However, these approaches are constrained by training time and memory. To tackle these shortcomings, we introduce three approaches starting with Multi-Level Dense Neural Networks (ML-DNN). ML-DNN draws inspiration from Multi-Level Monte-Carlo to efficiently sample and perform hierarchical learning thereby providing substantial time savings compared to the classical Dense Neural Network (DNN). Next, we implement Tensor Neural Networks, a quantum-inspired architecture that provides significant parameter savings and faster convergence while attaining the same accuracy as compared to a DNN. Finally, we introduce a model-based Reinforcement Learning algorithm which addresses the COD and is independent of the PDE family. We benchmark these models on parabolic PDEs, empirically showing their advantages over the current state-of-the-art models.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.003 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.000 | 0.001 |
| Scholarly communication | 0.001 | 0.001 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.003 |
| Insufficient payload (model declined to judge) | 0.003 | 0.001 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".