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Integration Matters for Learning PDEs with Backwards SDEs
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Identifies Euler-Maruyama discretization bias in one-step BSDE self-consistency losses for PDEs and proposes a Stratonovich BSDE formulation integrated with stochastic Heun, which removes the bias and matches PINNs on high-dimensional benchmarks.
Problem
BSDE-based deep learning solvers for high-dimensional PDEs empirically underperform PINNs; the root cause of this gap is unclear.
Approach
Interpret the forward and backward SDEs in Stratonovich form and discretize with stochastic Heun integration, so the one-step self-consistency BSDE loss has no non-vanishing discretization bias (unlike Euler-Maruyama), allowing single-step losses without interpolation hyperparameters.
Key results
- On 100D BSB, Heun-BSDE achieves RL2 .0535 vs .3735 for EM-BSDE in the full algorithm setting.
- On 100D HJB, Heun-BSDE achieves RL2 .0493 vs .3626 for EM-BSDE and .1260 for PINNs.
- Heun-BSDE performs similarly to FS-PINNs across cases; on 10D BZ Heun-BSDE gets .0228 vs .0351 for FS-PINNs.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Replace one-step Euler-Maruyama integration with stochastic Heun integration | BSDE solver · 100D HJB, full algorithm | RL2: .3626\pm.0113 -> .0493 \pm .0109 | Helped |
| Replace one-step Euler-Maruyama integration with stochastic Heun integration | BSDE solver · 100D BSB, full algorithm | RL2: .3735 \pm .0470 -> .0535 \pm .0113 | Helped |
| Replace one-step Euler-Maruyama integration with stochastic Heun integration | BSDE solver · 100D BZ, full algorithm | RL2: 3.1259 \pm .1807 -> 3.5619 \pm .2716 | Hurt |
| Replace one-step Euler-Maruyama integration with stochastic Heun integration | BSDE solver · 10D BZ, full algorithm | RL2: .1903 \pm .0066 -> .0228 \pm .0016 | Helped |
| Replace one-step Euler-Maruyama integration with stochastic Heun integration | BSDE solver · 100D BZ, batched algorithm | RL2: 5.7330 \pm .2342 -> 1.7973 \pm .1108 | Helped |
| Use forward-SDE trajectory sampling instead of the standard PINNs collocation sampling distribution | PINNs · 100D HJB, full algorithm | RL2: .1260\pm.0107 -> .0737 \pm.0110 | Helped |
| Increase the multi-step skip length | EM-BSDE · 10D BSB | RL2 | Mixed |
| Increase the multi-step skip length | Heun-BSDE · 10D BSB | RL2 | Hurt |
| Reduce the discretization step size | one-step EM-BSDE · 10D BSB | RL2 | No clear effect |
| Use the batched, sub-sampled rollout algorithm instead of the full rollout | FS-PINNs, EM-BSDE, and Heun-BSDE · PDE benchmarks | training time | Helped |
| Use stochastic Heun integration instead of Euler-Maruyama integration | batched BSDE solver · PDE benchmarks | training time: approximately 6x slower | Hurt |
Surprising findings
- Plain PINNs with the standard sampling measure performs much worse than FS-PINNs that sample collocation points from forward SDE trajectories.
- Finer step sizes do not close the EM-vs-Heun gap, since the EM bias is the same order as the PDE residual term.
Limitations
- Heun-BSDE has roughly 6x training overhead over EM-BSDE and needs float64 for stability.
- Theory only covers one-step and full-horizon extremes; intermediate skip lengths are studied empirically.
- Does not yet significantly outperform the best PINNs method; high-dimensional fully-coupled BZ 100D fails to converge.
Experimental setup
Benchmarks: 100D HJB 100D BSB 100D BZ 10D BZ pendulum swing-up optimal control
Metrics: relative L2 error (RL2)
Baselines: PINNs FS-PINNs EM-BSDE EM-BSDE (NR)
Builds on: PINNs self-consistency BSDE loss interpolating BSDE loss deep BSDE method
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