ARTEMIS: A Neuro Symbolic Framework for Economically Constrained Market Dynamics

Abstract

Deep learning models in quantitative finance often operate as black boxes, lacking interpretability and failing to incorporate fundamental economic principles such as no-arbitrage constraints. This paper introduces ARTEMIS (Arbitrage-free Representation Through Economic Models and Interpretable Symbolics), a novel neuro-symbolic framework combining a continuous-time Laplace Neural Operator encoder, a neural stochastic differential equation regularised by physics-informed losses, and a differentiable symbolic bottleneck that distils interpretable trading rules. The model enforces economic plausibility via two novel regularisation terms: a Feynman-Kac PDE residual penalising local no-arbitrage violations, and a market price of risk penalty bounding the instantaneous Sharpe ratio. We evaluate ARTEMIS against six strong baselines on four datasets: Jane Street, Optiver, Time-IMM, and DSLOB (a synthetic crash regime). Results demonstrate ARTEMIS achieves state-of-the-art directional accuracy, outperforming all baselines on DSLOB (64.96%) and Time-IMM (96.0%). A comprehensive ablation study confirms each component's contribution: removing the PDE loss reduces directional accuracy from 64.89% to 50.32%. Underperformance on Optiver is attributed to its long sequence length and volatility-focused target. By providing interpretable, economically grounded predictions, ARTEMIS bridges the gap between deep learning's power and the transparency demanded in quantitative finance.

Figures (6)

• Figure 1: Architecture of ARTEMIS. The framework processes irregularly sampled market data $\{(\mathbf{x}_i, t_i)\}_{i=1}^{N}$ through four tightly coupled modules. Module 1 (Laplace Neural Operator) encodes the input directly in continuous time via a learnable Laplace-domain kernel $\hat{\kappa}(\omega) = \sum_k \mathbf{A}_k/(\omega - \lambda_k)$, eliminating the need for interpolation or regular resampling. Module 2 (Neural SDE Latent Dynamics) evolves the encoded state under a stochastic differential equation $\mathrm{d}\mathbf{z} = \boldsymbol{\mu}_\theta(\mathbf{z},t)\,\mathrm{d}t + \boldsymbol{\sigma}_\varphi(\mathbf{z},t)\,\mathrm{d}\mathbf{W}$, where drift $\boldsymbol{\mu}_\theta$ and diffusion $\boldsymbol{\sigma}_\varphi$ are neural networks trained with two physics-informed penalties: a Feynman–Kac PDE residual $\mathcal{L}_{\mathrm{PDE}}$ that enforces local no-arbitrage conditions via an auxiliary pricing network $V_\psi$, and a market-price-of-risk penalty $\mathcal{L}_{\mathrm{MPR}}$ that bounds the instantaneous Sharpe ratio $\|\boldsymbol{\sigma}_\varphi^{-1}\boldsymbol{\mu}_\theta\|^2 \leq \kappa^2$ to economically plausible values. Module 3 (Symbolic Bottleneck) distils the latent dynamics into a sparse, human-readable combination of basis functions $\hat{y}_s = \sum_k w_k f_k(\mathbf{x})$ via a two-phase teacher–student procedure with Gumbel-Softmax selection, providing inherent interpretability without post-hoc approximation. Module 4 (Conformal Allocation) wraps predictions in distribution-free intervals $[\hat{y} \pm q_{1-\alpha}]$ via adaptive conformal prediction, and optionally solves a differentiable Kelly criterion portfolio problem. All components are trained jointly under the composite objective $\mathcal{L}_{\mathrm{total}} = \mathcal{L}_{\mathrm{forecast}} + \lambda_1\mathcal{L}_{\mathrm{PDE}} + \lambda_2\mathcal{L}_{\mathrm{MPR}} + \lambda_3\mathcal{L}_{\mathrm{consist}}$, with gradients backpropagated through the Euler–Maruyama SDE solver via the reparametrisation trick. The consistency loss $\mathcal{L}_{\mathrm{consist}}$ anchors the SDE trajectory to the encoder outputs at each time step, preventing latent drift.
• Figure 2: Temporal profiles of drift magnitude $\|\boldsymbol{\mu}(\mathbf{Z}, t)\|$ and diffusion magnitude $\|\boldsymbol{\sigma}(\mathbf{Z}, t)\|$ evaluated across the $100$-timestep input window on the DSLOB crash-regime test set. At each normalised time $t \in [0,1]$, the norms are computed over the full latent dimension and averaged across $256$ test samples; shaded bands denote $\pm 1\sigma$ across samples. Two observations are of economic significance. First, the diffusion magnitude $\|\boldsymbol{\sigma}\|$ increases monotonically toward the end of the sequence window, indicating that the model assigns growing uncertainty to more recent LOB states — consistent with the stylised fact that price impact and volatility are highest in the final moments before a regime transition. Second, the drift magnitude $\|\boldsymbol{\mu}\|$ exhibits a non-monotone profile with a mid-sequence peak, reflecting the model's learned representation of momentum followed by mean-reversion dynamics. Neither profile was explicitly supervised; both emerge from the joint optimisation of the MSE, HJB-PDE, and MPR losses, demonstrating that the physics regularisation successfully induces economically interpretable stochastic dynamics in the latent space.
• Figure 3: Vector field of the learned SDE dynamics projected onto the first two principal components (PC1–PC2) of the ARTEMIS latent space, evaluated at three canonical normalised times: $t = 0.1$ (early sequence), $t = 0.5$ (mid-sequence), and $t = 0.9$ (late sequence). The figure is arranged as a $2 \times 3$ grid. The top row shows drift quiver plots: each arrow represents the direction and magnitude of $\boldsymbol{\mu}(\mathbf{Z}, t)$ projected onto the PC1–PC2 plane via $\boldsymbol{\mu}_{\mathrm{PC}} = \boldsymbol{\mu} \mathbf{V}_{1:2}^\top$, where $\mathbf{V}_{1:2}$ are the top two PCA eigenvectors; arrow colour encodes drift speed $\|\boldsymbol{\mu}_{\mathrm{PC}}\|$. The bottom row shows diffusion heatmaps: the background colour encodes $\|\boldsymbol{\sigma}(\mathbf{Z}, t)\|$ computed directly in the full $64$-dimensional latent space at each grid point, with brighter regions indicating higher local volatility. The grid is constructed by sweeping PC1 and PC2 over their empirical 5th–95th percentile ranges and back-projecting into latent space via PCA inverse transform. At $t = 0.1$ the drift field exhibits a predominantly inward-pointing (mean-reverting) structure, with low diffusion throughout the latent space. By $t = 0.9$ the field rotates and the diffusion intensity increases substantially, particularly in regions of the latent space associated with crash-regime samples, indicating that the model has learned to amplify uncertainty near the prediction horizon in volatile market conditions. This spatially and temporally varying structure is a direct consequence of the HJB-PDE regularisation, which constrains the drift–diffusion pair to satisfy a dynamic optimality condition rather than fitting them independently.
• Figure 4: Performance degradation across the three DSLOB market regimes for all six benchmark models. The xx x-axis progresses from the training distribution (Normal, low volatility) through the validation distribution (Stress, medium volatility) to the held-out test distribution (Crash, high volatility with downward drift), representing a controlled out-of-distribution evaluation. ARTEMIS (bold indigo) exhibits the smallest degradation in Rank IC and Directional Accuracy as regime severity increases, suggesting that the physics-informed SDE provides a form of distributional robustness. Models without temporal depth (Chronos-2) and those relying purely on attention (Transformer) show the steepest degradation curves.
• Figure 5: Predicted versus actual mid-price return scatter plots for all six benchmark models evaluated on the DSLOB crash-regime test set. Each panel displays 2,000 randomly sampled predictions. The dashed diagonal represents the identity line (perfect prediction). RMSE and Rank IC are annotated in each title. ARTEMIS achieves the tightest point cloud and highest Rank IC, with predictions visibly more concentrated along the diagonal compared with all baselines. Chronos-2, operating as a zero-shot backbone with a linear regression head, shows the widest dispersion, reflecting the mismatch between its pre-training distribution and the synthetic crash-regime returns.