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From Biased to Unbiased Dynamics: An Infinitesimal Generator Approach
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Learns eigenfunctions and eigenvalues of the Langevin infinitesimal generator from biased simulations via a resolvent-based energy-regularized loss with a neural network dictionary, with unbiasing reweighting by exp(beta V).
Problem
Estimating the slow eigenfunctions and eigenvalues of the unbiased overdamped Langevin dynamics when only data from biased (enhanced-sampling) simulations are available; transfer operator methods require long lag times and reweighting that is only approximately valid.
Approach
Uses the infinitesimal generator and its resolvent with a regularized energy kernel, which lets reweighting by the Radon-Nikodym derivative be applied exactly to the energy (generator) risk, and trains a neural network dictionary to span the leading invariant subspace of the resolvent from biased samples.
Key results
- On Muller-Brown with metadynamics bias, the method's eigenvalue estimates are closer to ground truth than the Rayleigh-quotient generator learning baseline and it outperforms deepTICA on eigenfunctions.
- On alanine dipeptide Dataset 1 (only ~2 transitions), the first eigenfunction separates the two metastable states.
- Trained on alanine Dataset 2, the first eigenfunction is linearly related to the committor, matching a previously published committor model.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Learn the unbiased generator from biased simulations using the resolvent-based method instead of transfer-operator learning | Muller-Brown potential with metadynamics bias · Muller-Brown potential | eigenfunction estimation | Helped |
| Use the resolvent-based generator method instead of the prior generator-learning approach | Muller-Brown potential with metadynamics bias · Muller-Brown potential | eigenvalue estimation | Helped |
| Train the resolvent-based generator method on OPES data biased along suboptimal ψ | Alanine dipeptide · Alanine dipeptide Dataset 1 | transition-mechanism recovery | Helped |
| Train on the 50ns OPES dataset biased along φ | Alanine dipeptide · Alanine dipeptide Dataset 2 | committor correspondence | Helped |
| Use biased rather than unbiased trajectory data | Chignolin miniprotein · Chignolin folding/unfolding | agreement with reference trajectory | No clear effect |
Surprising findings
- Transfer operator reweighting is only approximately valid for lag time t>0, whereas the generator-based risk is exactly unbiased under reweighting.
Limitations
- Formulated only for time-homogeneous bias potentials.
- Requires knowledge of the bias V and diffusion coefficient beta (and ideally the potential U) for the unbiasing weights.
Experimental setup
Datasets: 1D double well biased simulations Muller-Brown metadynamics simulations alanine dipeptide OPES biased simulations (800 ns and 50 ns) chignolin 1 us deep-TDA biased simulation D.E. Shaw chignolin unbiased trajectory (106 us)
Benchmarks: 1D double well Muller-Brown potential alanine dipeptide chignolin miniprotein
Metrics: eigenvalue error eigenfunction sine distance qualitative eigenfunction shape
Baselines: deepTICA generator learning with Rayleigh quotient (Zhang et al.) transfer operator ridge regression with reweighting committor model
Builds on: transfer operator learning generator learning via resolvent (Kostic et al.) deepTICA metadynamics OPES
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