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Learning force fields from stochastic trajectories

Anna Frishman, Pierre Ronceray

arXiv:1809.09650v3cond-mat.softcond-mat.stat-mech

TL;DR

Finite Brownian trajectories contain limited information about an otherwise spatially variable force field. The paper presents Stochastic Force Inference, which uses trajectory data to estimate forces through a finite basis. Its accuracy depends on balancing basis completeness against finite-data error, and its scope is bounded by the assumed dynamics.

  • Problem

    A finite trajectory contains finite information, so only a finite number of combinations of a spatially variable force field can be estimated.

    Source: §II · p. 3

  • Method

    Stochastic Force Inference projects the force field onto a finite basis of known functions and fits its coefficients from trajectory data.

    Source: §II · p. 3

  • Results

    With a basis selected to maximize statistically resolvable information, typical squared force-inference error scales as τ^-1/2 with trajectory duration.

    Source: Appendix D · p. 7

  • Takeaways & Limitations

    The method provides a framework for reconstructing force and diffusion fields and measuring entropy production from Brownian trajectories.

    Source: §IV · p. 9

    The analysis is scoped to time-independent force fields and white-in-time noise; for time-varying forces, SFI captures their average projection onto observed variables.

    Source: §IV · p. 9

Abstract

from arXiv · show

When monitoring the dynamics of stochastic systems, such as interacting particles agitated by thermal noise, disentangling deterministic forces from Brownian motion is challenging. Indeed, we show that there is an information-theoretic bound, the capacity of the system when viewed as a communication channel, that limits the rate at which information about the force field can be extracted from a Brownian trajectory. This capacity provides an upper bound to the system's entropy production rate, and quantifies the rate at which the trajectory becomes distinguishable from pure Brownian motion. We propose a practical and principled method, Stochastic Force Inference, that uses this information to approximate force fields and spatially variable diffusion coefficients. It is data efficient, including in high dimensions, robust to experimental noise, and provides a self-consistent estimate of the inference error. In addition to forces, this technique readily permits the evaluation of out-of-equilibrium currents and the corresponding entropy production with a limited amount of data.

I. THE INFORMATION CONTENT OF BROWNIAN TRAJECTORIES

Brownian dynamics acts as a noisy channel carrying information about the force field, with a finite capacity that limits how quickly the force can be inferred. This capacity separates information associated with dissipation from information associated with spatial structure and bounds entropy production.

I. THE INFORMATION CONTENT OF BROWNIAN TRAJECTORIES

  • The channel capacity C bounds the rate at which a Brownian trajectory conveys information about the force field.The dynamics can be viewed as transmitting force values along a trajectory through noise added to the observed motion.Source: §I · p. 2
  • For a constant force with isotropic diffusion, C = F^2/4D and the force becomes resolvable when the trajectory contains more than d units of information I.In this example, I = Cτ and the relative error of the end-to-end velocity estimator is d/2I.Source: §I · p. 2
  • The capacity decomposes into non-negative contributions associated with entropy production and the spatial structure of the steady-state distribution.The structural contribution, the inflow rate G, can be nonzero even at equilibrium.Source: §I · pp. 2–3
  • The non-negative inflow rate implies the general upper bound ˙S ≤ 4C for entropy production in Brownian systems.The inflow rate reflects information the force supplies to maintain probability gradients against diffusion.Source: §I · p. 3
  • The capacity also quantifies how quickly trajectories become distinguishable from force-free Brownian motion.It can be expressed as a trajectory-wise information gain about the force field.Source: §I · p. 3

II. STOCHASTIC FORCE INFERENCE

Stochastic Force Inference (SFI) estimates force fields by projecting them onto a finite basis and fitting its coefficients from trajectory data. Its finite-dimensional estimates can also be used to approximate currents and provide lower bounds on entropy production.

II. STOCHASTIC FORCE INFERENCE

  • SFI approximates the force by projecting it onto a finite basis of known functions using the steady-state distribution as the least-squares measure.A finite trajectory allows only a finite number of force-field combinations to be estimated.Source: §II · p. 3
  • Projection of phase-space velocity coefficients yields approximate currents and an entropy-production estimate that is a lower bound on total entropy production.When only some degrees of freedom are observed, the framework estimates forces averaged over hidden variables and bounds entropy production from observable currents.Source: §II · p. 4
  • The operational coefficient estimator is a discrete Itô integral, equivalent to linear regression of local force estimates, with typical squared relative error scaling as Nb/2Ĩb.The empirical information estimate Ĩb uses the inferred coefficients; resolving the coefficients requires trajectory information to exceed the number of inferred parameters.Source: §II · p. 4
  • 0.15 squared relative error on the force coefficients, versus the self-consistent estimate 0.16, was obtained for a 2D Ornstein-Uhlenbeck trajectory.The trajectory contained 27.6 bits of information, and the first-order polynomial basis had six degrees of freedom; the inferred force was extrapolated beyond the sampled path.Source: Figure 3 · p. 4, §II · p. 4

Appendix D)

The section develops ways to infer circulation, forces, and entropy production from stochastic trajectories, including noisy, nonlinear, and high-dimensional settings. It also explains how basis choice trades representation accuracy against overfitting.

Entropy production

  • The entropy-production estimator is biased by 2Nb/τ, but its error can be self-consistently controlled from a single trajectory when several kB per degree of freedom have been dissipated.The estimate concerns entropy production in the observed channel and currents.Source: Appendix D · p. 5

6D circulating Ornstein-Uhlenbeck processes

  • In a 6D circulating Ornstein–Uhlenbeck process, SFI identifies the circulation plane and force field, while angular error falls with trajectory length.With measurement noise, inferred entropy production underestimates the exact value because currents are blurred; observing only a projection captures only observable entropy production.Source: Appendix D · p. 5, Figure 4 · p. 5

Measurement noise

  • Time-reversal antisymmetry makes velocity coefficients robust to measurement noise, enabling an unbiased force estimator from the Itô-to-Stratonovich relation.The modified estimator requires a smooth projection basis and known diffusion tensor; spatial blurring and finite time steps can still bias force or entropy-production estimates.Source: Appendix D · p. 6

Nonlinear force fields

  • For nonlinear forces, expanding the projection basis captures finer spatial features and increases the captured capacity and entropy production, but can eventually overfit.A linear basis can capture covariance and current circulation while missing finer force structure.Source: Appendix D · p. 6, Figure 5 · p. 6

Basis size and inference error

  • With finite trajectories, force error first falls and then rises as basis size increases, reflecting the trade-off between underfitting and fitting noise.The proposed selection criterion maximizes inferred information minus one standard deviation; typical squared force error scales as τ^-1/2, or τ^-1 when the basis exactly represents the force.Source: Appendix D · pp. 6–7, Figure 6 · p. 7

Nonlinear force fields

  • For nonlinear force models, Figure 5 compares inferred with exact forces and bootstrapped trajectories, and plots capacity and entropy production against basis size.The figure includes polynomial and Fourier bases and a coarse-grained approximation.Source: Figure 5 · p. 6

Active interacting particles

  • For identical interacting particles, exchange symmetry reduces the parameters to infer, so many particles can enhance statistics and support accurate force and pair-interaction inference from limited data.The method does not require an equilibrium assumption for pair interactions.Source: Appendix D · p. 7, Figure 7 · p. 7

III. INHOMOGENEOUS DIFFUSION

This section extends inference to state-dependent diffusion by estimating diffusion and drift fields from trajectories, then combining them to recover physical forces. Noise correction is available for diffusion estimates, while force recovery requires an adequate approximation of the diffusion divergence.

III. INHOMOGENEOUS DIFFUSION

  • Physical force recovery from inhomogeneous diffusion requires inferring both diffusion and drift, and depends on adequately approximating the diffusion divergence.The drift and physical force differ by a term involving the divergence of diffusion.Source: §III · pp. 8–9

Diffusion inference

  • The diffusion field is approximated in a function basis and estimated from trajectory averages using local displacement products; its error vanishes for long trajectories in the tested models.Tests include a one-dimensional ratchet and a two-dimensional harmonic trap with a constant diffusion gradient.Source: §III · p. 8, Figure 8 · p. 8

Measurement-noise correction

  • The Eq. 12 diffusion estimator is biased by measurement noise and can become ineffective when that noise exceeds typical displacements; a bias-corrected local estimator addresses the bias.The correction increases relative error for short trajectories.Source: §III · p. 8

Drift inference

  • The drift projection estimator uses an Itô-to-Stratonovich conversion to avoid measurement-noise bias, and its convergence resembles that of the Itô estimator.The direct Itô estimator is impractical for experimental data because even moderate measurement noise induces large coefficient errors.Source: §III · pp. 8–9

Force reconstruction

  • Combining diffusion and drift projection estimators with Eq. 11 yields an estimator for the physical force field.Figure 8 demonstrates inference of diffusion and force for a one-dimensional ratchet and a two-dimensional equilibrium trap with a diffusion gradient.Source: Figure 8 · p. 8, §III · p. 9

IV. DISCUSSION

The discussion presents SFI as a data-efficient approach to reconstructing force and diffusion fields from Brownian trajectories, with strong performance in a noisy single-molecule benchmark. It also emphasizes the information limits of inference, the method’s scope, and its potential use across stochastic systems.

IV. DISCUSSION

  • SFI reconstructs force and diffusion fields by fitting force to a linear combination of known functions, using the information available per degree of freedom.The paper benchmarks accuracy on model systems using data comparable to current experiments.Source: §IV · p. 9
  • SFI outperforms InferenceMAP and adaptive grid binning on drift and diffusion errors in the 2D benchmark, and is the only unbiased drift estimator with measurement noise.The comparison evaluates ideal and noisy data across different numbers of trajectories.Source: Figure 9 · p. 9, §IV · p. 9
  • SFI also estimates entropy production from currents and performs well in high dimensions with short, noisy trajectories and inhomogeneous diffusion.The discussion contrasts this scope with alternative entropy-production methods that require large amounts of well-resolved data and are limited to relatively low dimensions and homogeneous diffusion.Source: §IV · p. 9
  • The framework is scoped to time-independent forces and white-in-time noise; for non-Markovian dynamics such as colored noise, force inference is complicated, though phase-space-velocity projection remains valid.For time-varying forces, SFI captures their average projection onto observed variables.Source: §IV · pp. 9–10
  • The discussion positions the framework for stochastic systems modeled by Brownian dynamics, including biological, climate, and financial applications.The authors also suggest combining force fitting with sparsity-promoting techniques to identify simple dynamical rules.Source: §IV · p. 10

Material and methods

Appendix A: Gaussian channel interpretation of Brownian dynamics

  • Finite-rate sampling loses negligible information when the information per sample is small, so increasing acquisition frequency then yields little additional force-field information.The paper notes this matters when acquisition rate trades off against experiment duration, for example because of photobleaching.Source: Appendix A · p. 13

1. The capacity as a Kullback-Leibler divergence rate

  • For inhomogeneous diffusion, the paper uses a trajectory-based capacity definition related to the Kullback–Leibler divergence rate between forced and same-diffusion zero-force trajectories.For constant diffusion, this reduces to the capacity discussed for Brownian dynamics; the transmission-theory interpretation is more complicated with multiplicative noise.Source: §1 · p. 14

2. The inflow rate

The inflow rate measures the information needed to maintain spatial structure against diffusion and can remain positive at equilibrium. The paper relates it to force-field divergence and presents an alternative decomposition of capacity into dissipation and inflow.

2. The inflow rate

  • The paper generalizes the inflow rate to inhomogeneous diffusion and discusses its trajectory-based interpretation.Source: §2 · pp. 14–15
  • The inflow rate is nonzero even at equilibrium, where it measures steady-state spatial heterogeneity rather than irreversible currents.The paper describes it as information the force continuously injects to maintain spatial structure.Source: §2 · p. 15
  • The inflow rate can be written as minus the average divergence of the force, but in equilibrium overdamped Brownian dynamics it does not represent actual entropy production.The paper contrasts this with deterministic systems, where phase-space contraction corresponds to mean environmental entropy production.Source: §2 · p. 15

3. Different decompositions of the capacity and the relation to traffic

  • A second capacity decomposition separates a time-antisymmetric entropy-production term from a time-symmetric inflow-rate term.The paper relates this trajectory-based decomposition to dynamical entropy and traffic.Source: §3 · p. 16

Appendix C: Stochastic Force Inference: estimating Fµα and its error

Stochastic Force Inference estimates force projections from trajectory averages, then approximates the unknown phase-space measure with an empirical one. The resulting empirical basis converges toward the phase-space basis as trajectory duration grows.

1. The force as a trajectory average

  • SFI estimates force-projection coefficients as averages along a trajectory, making them estimable from finite trajectories.The force is expressed using measurable trajectory quantities and projected onto phase-space basis functions.Source: §1 · pp. 16–17

2. Projection on the empirical basis

  • Because the phase-space measure is unknown, SFI approximates it with the empirical measure and constructs empirical projectors from that measure.The empirical measure is obtained from a time average along the trajectory.Source: §2 · p. 17
  • For typical trajectories, empirical projectors converge toward phase-space projectors as O(sqrt(τ0/τ)), where τ0 is a relaxation time.For a polynomial basis, convergence at order n is related to convergence of the n-th cumulant.Source: §2 · p. 17

3. Statistics of the error in the inference of the projection coefficients

The section analyzes systematic bias and fluctuating error in estimated force-projection coefficients, accounting for convergence of the empirical basis and dimensionless scaling across coordinates.

3. Statistics of the error in the inference of the projection coefficients

  • Error moments are evaluated in dimensionless coordinates because phase-space coordinates can have different physical dimensions.The rescaled error coordinates have dimension t^-1/2.Source: §3 · p. 18
  • The systematic bias decays as O(1/τ), faster than the fluctuating inference error, which scales as O(τ^-1/2).The bias bound arises from correlations between noise and fluctuations in the empirical orthonormalization matrix.Source: §3 · p. 18
  • The fluctuation calculation includes a subleading remainder associated with convergence of the empirical projected basis to its long-trajectory limit.The derivation bounds this remainder using typical trajectory fluctuations of the empirical orthonormalization matrix.Source: §3 · pp. 18–19

4. Self-consistent estimate of the error on the projected force

SFI provides a self-consistent estimate of projected-force inference error using trajectory-dependent quantities, and its force estimator is also a maximum-likelihood estimator.

4. Self-consistent estimate of the error on the projected force

  • The empirical information estimator is biased by Nb/2 at leading order.The passage also states that its variance is well approximated, but the supplied text does not provide the expression.Source: §4 · p. 19
  • The squared typical error on force-projection coefficients can be estimated from trajectory-dependent quantities, assuming the diffusion matrix is known.The relative inference error is estimated as Nb/(2Î).Source: §4 · p. 20

5. The force estimator and maximum likelihood

  • The proposed force estimator maximizes the trajectory log-likelihood because the empirical projectors are independent of the force coefficients being estimated.The estimator solves the likelihood condition for those coefficients.Source: §5 · p. 20

Appendix D: Inference of velocities and entropy production

The appendix describes inference of phase-space velocities and entropy production from trajectories, including projected estimates and their limitations. With incomplete observations, the projected quantities remain lower bounds on total entropy production and capacity.

Appendix D: Inference of velocities and entropy production

  • Projected heat-related entropy production is not positive definite unless the entire force is resolved, and projected heat and total entropy production generically differ.Thus, the heat-related projection does not provide a bound on entropy production.Source: Appendix D · p. 22
  • Projected total entropy production is positive definite and provides a lower bound on entropy production.The appendix also describes estimating phase-space currents or velocities and entropy production from a trajectory.Source: Appendix D · pp. 21–22
  • The phase-space velocity estimator has two sources of error, and the authors lack a formal estimate for one term.The resulting error estimate is described as order-of-magnitude rather than fully rigorous.Source: Appendix D · p. 22

Appendix E: Incomplete observations and time-dependent forces

  • For incomplete observations, SFI projects the force onto observed variables, effectively averaging over hidden variables, and the projected entropy production and capacity remain lower bounds on their totals.The result applies when the full system follows a constant-force Langevin equation.Source: Appendix E · p. 23

Appendix F: Inference with imperfect data: measurement noise and time discretization

The appendix develops estimators for forces, currents, diffusion, and drift when measurements are noisy or diffusion varies in space. It also identifies how measurement noise and time discretization limit accuracy and help determine data-acquisition settings.

Appendix F: Inference with imperfect data: measurement noise and time discretization

  • Velocity projection estimates are unaffected by time-uncorrelated measurement noise apart from a zero-mean fluctuating term, while the proposed force estimator uses the relation between Itô and Stratonovich integration.The force-estimation treatment remains incomplete: empirical information and entropy production are biased, and large measurement noise relative to trajectory variation adds bias.Source: Appendix F · p. 24
  • Force changes within a time step blur the inferred field and produce a discretization bias; Stratonovich averaging reduces the stated squared error by a factor of 4.The discretization error estimate can be compared with the limited-information error to assess which source constrains inference.Source: Appendix F · pp. 24–25
  • Comparing discretization error with finite-trajectory error helps determine whether trajectory length or frame rate limits inference, informing acquisition when photobleaching limits captured frames.The appendix also warns that large time steps blur force variations and that measurement-noise biases can require long trajectories and not-too-small time steps.Source: Appendix F · pp. 24–25

Appendix G: Inference in the presence of an inhomogeneous diffusion coefficient

  • For state-dependent diffusion, the appendix follows the force-inference logic, defining local diffusion estimates and projections to construct an estimator.With noisy measurements, it proposes a modified diffusion estimator and uses it in a drift-projection estimator adapted to measurement noise.Source: Appendix G · p. 25

1. Estimate of the error on the projected diffusion coefficient

The appendix estimates errors in projected diffusion and drift, including with measurement noise. It describes a trade-off between biased and corrected diffusion estimators and limits error control for physical-force estimates when diffusion varies in space.

1. Estimate of the error on the projected diffusion coefficient

  • Under bounded diffusion, the correction from estimated rather than exact projectors is sub-leading, of order O(τ^-3/2).The error derivation assumes the diffusion tensor is bounded in the domain.Source: §1 · p. 26

2. Inference of the diffusion coefficient with measurement noise

  • For noisy diffusion inference, the corrected estimator’s measurement-noise error decreases with trajectory length, while the uncorrected estimator’s bias remains; the corrected estimator has four times larger squared error from signal contributions.For short trajectories with sufficiently small measurement noise, the uncorrected estimator may perform better; its error eventually saturates as the corrected estimator continues decreasing.Source: §2 · p. 27

3. Drift inference for an inhomogeneous diffusion coefficient

  • For inhomogeneous diffusion, drift projection uses the relation between Itô and Stratonovich integrals and a modified diffusion estimator to correct measurement-noise bias.With ideal data, the appendix recommends the less computationally complex estimator described in Eq. G15.Source: §3 · p. 27

4. Estimate of the error on the projected drift

  • The appendix cannot provide an error estimate for the inferred physical force because the error in the gradient of the inferred diffusion is uncontrolled without regularity assumptions.The force estimate combines inferred drift with the divergence of the inferred diffusion tensor.Source: §4 · p. 28

1. Overdamped Langevin simulations

The simulations test Stochastic Force Inference across Brownian models, including nonlinear and high-dimensional systems, active particles, and spatially varying diffusion. Results show both accurate recovery in selected settings and important dependence on basis choice and measured trajectory information.

1. Overdamped Langevin simulations

  • In the 2D Ornstein–Uhlenbeck example, the actual force-coefficient error falls within the self-consistent confidence interval Nb/2Ĩb = 0.16.The setup uses a linear-trap process and a first-order polynomial basis; the passage reports an inferred information content of 27.6 bits.Source: §2 · pp. 28–29

4. Nonlinear obstacle process (Figure 5A, C, E, G, I)

  • The nonlinear obstacle force cannot be captured exactly by the chosen basis, and the simulations compare polynomial orders and basis families for projected capacity and entropy production.The reported basis comparison uses a long trajectory and includes polynomial, Fourier, and grid coarse-graining choices.Source: §4 · p. 30

5. Lorenz process (Figure 5B, D, F, H, J)

  • For the Lorenz process, polynomial orders 2 and 3 capture the force field, while order 2 captures all capacity but not all entropy production.The remaining entropy-production difference is attributed to nonzero exchange terms with higher-order moments.Source: §5 · p. 31

6. Active Brownian Particles simulations (Figure 7)

  • For active-particle interactions, the symmetrized projection shares coefficients across particles, reducing approximation degrees of freedom; polynomial distance bases emphasize far-field interactions.The appendix notes that decaying functions of distance are better adapted to interactions that decay with distance, and SFI does not optimize shape parameters.Source: §6 · pp. 31–32

7. One-dimensional ratchet process (Figure 8A-D)

  • The one-dimensional ratchet simulation applies SFI to infer a periodic force and spatially varying diffusion, then examines field convergence as trajectory duration increases.The simulation uses an adapted trigonometric basis for both fields.Source: §7 · p. 32

8. Minimal 2D model with diffusion gradient (Figure 8E-H)

  • In the 2D equilibrium model with a diffusion gradient, the appendix reports convergence of the diffusion estimator with trajectory length, including runs with added measurement noise.The chosen mobility and force leave the probability distribution unaffected by the diffusion inhomogeneity.Source: §8 · p. 33

9. Reconstruction of the drift and diffusion field for a complex 2D process (Figure 9)

The section compares stochastic force inference (SFI), grid binning, and InferenceMAP on simulated drift and diffusion fields, including noisy trajectories. SFI provides the strongest drift inference, while grid binning converges more slowly with ideal data and fails to converge with noisy data.

9. Reconstruction of the drift and diffusion field for a complex 2D process (Figure 9)

  • The comparison simulates a cellular diffusion process with spatially varying isotropic diffusion and a drift combining a harmonic trap, torque, and four Gaussian traps.Trajectories use exact data or coordinates blurred by Gaussian white noise of amplitude 0.1.Source: §9 · p. 33
  • The study uses 4 to 10^4 independently generated, 100-step trajectories, each starting at steady state and carrying an average of 2.8 bits of drift-field information.Inference accuracy is assessed by normalized mean-squared error between exact and inferred fields along trajectories.Source: §9 · p. 33
  • Grid binning estimates a constant drift vector and diffusion coefficient per regular bin, with adaptive bin size increasing spatial resolution and per-bin accuracy as data accumulate.The method uses maximum-likelihood estimators within each bin.Source: Maximum-likelihood grid binning · p. 33
  • With ideal data, grid binning converges to exact fields, but more slowly than SFI; InferenceMAP performs slightly worse than SFI on diffusion inference.InferenceMAP's drift inference does not exceed grid binning, and SFI significantly outperforms it on drift inference.Source: Maximum-likelihood grid binning · p. 33, InferenceMAP · p. 34
  • On noisy trajectories, SFI quantitatively matches the exact drift, while grid binning and InferenceMAP are biased and overestimate it.Grid binning becomes biased and does not converge with noisy data; SFI significantly outperforms InferenceMAP on drift inference.Source: Maximum-likelihood grid binning · p. 33, Figure 10 · p. 34, InferenceMAP · p. 34
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