Reflected Event-Driven Residual Dynamics
Implementation & benchmark of arXiv:2607.29548 — Generalized reflected BSDEs with irregular obstacles driven by RCLL increasing processes on general filtered space
Source paper: Generalized reflected BSDEs with irregular obstacles driven by RCLL increasing processes on general filtered space arXiv:2607.29548 ⓘ · analyzed Aug 31, 2026
AI-generated research hypothesis, automatically tested. Not peer-reviewed.
Idea description
Replace a uniformly discretized recurrent or continuous-depth model with hybrid hidden-state dynamics: integrate a learned drift between event times, then apply a one-sided reflection update at each irregular observation or constraint event. The reflection prevents the hidden state from violating a lower obstacle, while the explicit jump decomposition avoids smearing abrupt information changes across many small residual steps.
Formulas
Mathematical statement
The paper decomposes the state process into an RCLL component and right-jump corrections: $Y^{n}_{t}=Y^{n,\ast}_{t}+\sum_{t\leq s<T}\Delta_{+}Y^{n}_{s}$, where $Y^{n}$ is the full state, $Y^{n,\ast}$ is the right-continuous component, and $\Delta_{+}Y^{n}_{s}=Y^{n}_{s+}-Y^{n}_{s}$ is the right jump. At event times $\sigma_{n,i}$, the obstacle correction is $\sum_{t\leq s<T}\Delta_{+}Y^{n}_{s}:=-\sum_{t\leq\sigma_{n,i}<T}(Y^{n}_{\sigma_{n,i}+}-\xi_{\sigma_{n,i}})^{-}$, where $\xi_{\sigma_{n,i}}$ is the lower obstacle and $x^{-}=\max(-x,0)$. The penalized equation contains $n(Y^{n}_{s-}-\xi_{s-})^{-}ds$, with $n>0$ controlling continuous violation penalties. We adapt this into a forward hybrid recurrence: drift updates model ordinary evolution, event jumps inject observations, and $h\leftarrow h+\lambda(\xi-h)_{+}$ implements soft reflection.
Implementation notes
Integrate this at the hidden-state update of an irregular-time RNN, neural CDE, or state-space model. For each sequence, sort observation times $t_0<\cdots<t_K$; let $h_k\in\mathbb{R}^d$ be the hidden state, $u_k$ the observation embedding, and $\Delta t_k=t_{k+1}-t_k$. First perform continuous evolution with $h_{k+1}^{\mathrm{pre}}=h_k+\Delta t_k f_\theta(h_k,u_k,\Delta t_k)$. At an observation event, add a jump encoder: $h_{k+1}^{\mathrm{jump}}=h_{k+1}^{\mathrm{pre}}+g_\phi([u_{k+1},\Delta t_k])$; when no event occurs, use zero jump. Define an obstacle vector $\xi_{k+1}=q_\psi(u_{k+1})$ or a fixed task-specific lower bound. Apply the differentiable penalty $h_{k+1}=h_{k+1}^{\mathrm{jump}}+\lambda_k(\xi_{k+1}-h_{k+1}^{\mathrm{jump}})_+$, or hard componentwise reflection $h_{k+1}=\max(h_{k+1}^{\mathrm{jump}},\xi_{k+1})$. Pseudocode is: h=h0; for k: hpre=h+dt[k]*f(h,u[k],dt[k]); hpre=hpre+event[k]*g(u[k+1],dt[k]); violation=relu(xi[k+1]-hpre); h=hpre+lambda[k]*violation. The paper supplies the drift/jump/reflection structure; $f,g,q$ and $\lambda_k$ are learned or selected empirically. Begin on PhysioNet irregular-time classification, comparing against GRU-D and Neural CDE baselines at equal hidden width and FLOPs. Measure validation loss, calibration, obstacle violations, and robustness after randomly deleting observations. Success means fewer violations and better missing-event robustness at equal compute, or equal accuracy with fewer solver evaluations.
Verification
This idea has not been verified yet.
Verification happens in two stages: Stage 1 — a mechanism check on a toy system confirms the claimed mathematical phenomenon reproduces; Stage 2 — a benchmark implements the idea on a real (small) neural network task and compares it against a tuned baseline over 8 paired seeds with a permutation test.
Artifacts
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