Stable and Calibrated Latent Force ODE Filters

Paul Fischer; Jonathan Schmidt; Philipp Hennig

cs.LGOct 24, 2026

Stable and Calibrated Latent Force ODE Filters

Authors: Paul Fischer, Jonathan Schmidt, Philipp Hennig

Organizations: Tübingen AI Center University of Tübingen

Abstract

Estimating unobserved forces in mechanistic systems is a central task of scientific machine learning. Approximate Gaussian filtering methods that synthesize mechanistic and empirical information have shown remarkable capacity in jointly identifying latent processes together with the system state in a single forward-backward pass. The resulting latent force ODE filter framework channels information from observational data specifically into the unknown parts of the system. This work addresses open questions about calibration and stability of the method. We show that the joint posterior over state and force is the reweighted conventional latent force posterior. This uncovers a crucial dependence of the stability of the method on the prior variance of the system state, which effectively controls the flow of information from observations to the latent force through the mechanistic system. From this insight, we construct an improved algorithm that provides stable and calibrated plug-and-play inference. We demonstrate the improved framework on several simulated experimental settings.

Figures & tables

Figure 1: The state diffusion controls the latent force inference. Latent force ODE filter on the Duffing oscillator of Eq. ( 16 ) with the position observed (Appendix D ). Center: the diffusion

\sigma_{x}^{\star 2}

that maximizes

I(U;Y\mid Z)

(Sec. 3.4 ). Left and right:

10^{8}

times smaller and

10^{6}

times larger. Bands: mean

\pm 2

standard deviations of the classical latent force model with the probabilistic numerical solver,

\pi(X,U)

, and of the full latent force ODE filter posterior

p(X,U\mid Y,Z)

Panel of Figure 2 — Figure 2: Directed graphical model of the latent force ODE filter. State and force are coupled only through the residual pseudo-observation and the initial state. $\varphi_{x}$ , $\varphi_{u}$ : prior hyperparameters. Shaded: observed. Plates: $N$ grid nodes and $K\leq N$ observation times. : mechanism $f$ , replaced at inference by $Z$ .

Criterion $\mathcal{J}$	Definition	Passes
$\mathcal{L}_{\mathrm{OBS}}$	Eq. ( 8 )	1
$\log p(Y\mid Z)$	$\mathcal{L}_{\mathrm{ODE}}+\mathcal{L}_{\mathrm{OBS}}-\log p(Z)$	2
$D_{\xi}$	Eq. ( 14 )	2
$I_{\xi}(U;Y\mid Z)$	Eq. ( 13 )	4

Table 1: Criteria

\mathcal{J}

for the state diffusion (line 8 of Algorithm 1 ): evidence (top) and information (bottom), causal member first, with the cost of one evaluation in filter passes.

Figure 3: The force error is smallest where $I(U;Y\mid Z)$ is largest. (a) Relative force error and (b) mutual information $I(U;Y\mid Z)$ against the state diffusion $\sigma_{x}^{2}$ on Duffing (Appendix D ), with the IEKS and $\varphi_{u}$ at the truth. One line per grid density $\kappa$ (color bar). Per grid, marks the maximizer of $I(U;Y\mid Z)$ and that of $\mathcal{L}_{\mathrm{ODE}}$ .

Panel of Figure 5 — Algorithm 1 Latent force ODE filter with the diffusion selected by a criterion $\mathcal{J}$ (Table 1 ).

Figure 4: Error ratio, force

\chi^{2}

and roughness ratio of the criteria and the classical LFM on the

60

problems. Columns: Duffing, FitzHugh–Nagumo and Lorenz-96, one dot per problem and method. Top: error ratio, the relative force error over the smallest relative force error of any method on the same problem, with the miss threshold

1.5

dashed. Fits above it are outlined, also in the rows below. Middle: force

\chi^{2}

, with the 95 % interval of a calibrated posterior whose errors are correlated as the true force prior in gray (Appendix E.6 ). Bottom: roughness ratio

(\sigma_{u}^{2}/\ell_{u}^{5})/(\sigma_{u}^{\star 2}/\ell_{u}^{\star 5})

of the fitted force prior to the true one, below one for a force that is too smooth and above one for a force that is too rough.

Appendix figures & tables8 assets

Supplementary material from the paper’s appendix.

Appendix

Figure A.1: The

4

misses of

\mathcal{L}_{\mathrm{OBS}}

. Force (top) and the scored state with its data (bottom) against

t/T

. On Duffing the acceleration is observed, so no data are shown. Bands: mean

\pm 2

standard deviations. Purple (outline):

\sigma_{x}^{2}

\mathcal{L}_{\mathrm{OBS}}

. Blue:

\sigma_{x}^{2}

I(U;Y\mid Z)

. Navy (dotted): classical LFM. Dashed: truth.

Figure A.2: The

4

misses of

D

, all on Lorenz-96. Force (top) and the observed site

x_{0}

with its data (bottom) against

t/T

. Bands: mean

\pm 2

standard deviations. Purple (outline):

\sigma_{x}^{2}

D(U;Y\mid Z)

. Blue:

\sigma_{x}^{2}

I(U;Y\mid Z)

. Navy (dotted): classical LFM. Dashed: truth.

Figure A.3: The miss of

\log p(Y\mid Z)

(left) and of the classical model (right). Force (top) and the scored state (bottom) against

t/T

. Bands: mean

\pm 2

standard deviations. Purple (outline): the failing method. Blue:

\sigma_{x}^{2}

I(U;Y\mid Z)

. Navy (dotted): classical LFM. Dashed: truth.

Figure A.4: Error ratio and force

\chi^{2}

off the model. One dot per problem and method, for truth forces outside the model’s prior (pooled over Duffing, FitzHugh–Nagumo and Lorenz-96,

10

problems each) and for the SIR model. Top: error ratio, with the miss threshold

1.5

dashed. Fits above it are outlined, also in the row below. Bottom: force

\chi^{2}

, with the 95 % interval of a calibrated posterior whose errors are correlated as the model’s force prior in gray (Appendix E.6 ), one interval per pooled system, drawn on top of each other.

Figure A.5: Example posteriors off the model. One problem per kind of truth force (columns), chosen as in the text. Force (top) and the scored state with its data (bottom) against

t/T

. Bands: mean

\pm 2

standard deviations. Blue:

\sigma_{x}^{2}

I(U;Y\mid Z)

. Purple (outline):

\sigma_{x}^{2}

\mathcal{L}_{\mathrm{OBS}}

. Navy (dotted): classical LFM. Dashed: truth. The posteriors nearly coincide, and for the piecewise-constant force all of them round off the jumps, which a Matérn-

5/2

prior cannot represent.

	$\sigma_{x}^{2}\to 0$	$\sigma_{x}^{2}\to\infty$
$L_{x}$	$L_{x\mid u}(I-\Pi)$	$L_{x\mid u}$
$L_{u}$	$P_{uu}^{-}B^{\top}S_{u}^{+}\Pi$	$P_{uu}^{-}B^{\top}S_{x}^{-1}\to 0$
$\mu_{x}^{\mathrm{+}}$	$-L_{x\mid u}(I-\Pi)\hat{z}$	$-L_{x\mid u}\hat{z}$
$\mu_{u}^{\mathrm{+}}$	$-P_{uu}^{-}B^{\top}S_{u}^{+}\Pi\hat{z}$	$0$
$P_{xx}^{\mathrm{+}}$	$P_{xx}^{-}-L_{x\mid u}(I-\Pi)CP_{xx}^{-}\to 0$	$P_{x\mid u}^{\mathrm{+}}\to\infty$
$P_{xu}^{\mathrm{+}}$	$-P_{xx}^{-}C^{\top}\Pi^{\top}S_{u}^{+}BP_{uu}^{-}\to 0$	$-L_{x\mid u}BP_{uu}^{-}$

Appendix

Table C.1: Limits of the single update as

\sigma_{x}^{2}\to 0

and

\sigma_{x}^{2}\to\infty

. Leading order of the single residual update (top), of the innovation covariance and gains of the subsequent observation update (middle), and of the posterior after both updates (bottom), as

\sigma_{x}^{2}\to 0

and as

\sigma_{x}^{2}\to\infty

. Blue : the entry vanishes in the limit. Orange : it diverges in proportion to

\sigma_{x}^{2}

. Black: it tends to a finite limit that does not depend on the diffusion.

P_{xx}^{-}

S_{x}

and

P_{x\mid u}^{\mathrm{+}}

are proportional to

\sigma_{x}^{2}

. If the force reaches every component of the residual,

\Pi=I

, and

L_{x}

and the state means vanish in proportion to

\sigma_{x}^{2}

for

\sigma_{x}^{2}\to 0

Figure D.1: Relative error and $\chi^{2}$ of force and state against the state diffusion. Relative error of force and state (top rows) and their $\chi^{2}$ (bottom rows) against $\sigma_{x}^{2}$ on Duffing, FitzHugh–Nagumo and Lorenz-96 (columns) with $\varphi_{u}$ at the truth, one line per grid density $\kappa$ , computed with the IEKS iterated to convergence. Only converged points are drawn. Diamonds: the $\sigma_{x}^{2}$ maximizing $I(U;Y\mid Z)$ on each grid (Fig. D.2 ), placed on the curves by interpolation. Squares: the $\sigma_{x}^{2}$ maximizing $\mathcal{L}_{\mathrm{ODE}}$ . Both in the grid’s color.

Figure D.2: Force error and the four criteria against the state diffusion on three problems that $\mathcal{L}_{\mathrm{OBS}}$ misses. Relative force error, $\mathcal{L}_{\mathrm{OBS}}$ and $\log p(Y\mid Z)$ below their maxima, $D(U;Y\mid Z)$ and $I(U;Y\mid Z)$ (rows) against $\sigma_{x}^{2}$ on the first FitzHugh–Nagumo, the 11th Duffing and the 10th Lorenz-96 problem of Sec. 4.2 (columns), with the IEKS and $\varphi_{u}$ at the truth, one line per grid density $\kappa$ . Only converged points are drawn. Each criterion’s row marks its own maximizer: stars for $\mathcal{L}_{\mathrm{OBS}}$ , plus signs for $\log p(Y\mid Z)$ , triangles for $D(U;Y\mid Z)$ and diamonds for $I(U;Y\mid Z)$ .

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K. Kumar is with the School of Engineering and Applied Science, Ahmedabad University, India · K. Kumar and S. Särkkä are with the ELLIS Institute Finland and the Department of Electrical Engineering and Automation, Aalto University, Finland · S. Das is with the Department of Electrical Engineering, Indian Institute of Technology (BHU) Varanasi, India

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This paper proposes a Bayesian filtering-based approach for learning the dynamics of a physical system from partial, noisy measurements. We model the system dynamics using a Lagrangian mechanics formulation. As in Lagrangian neural networks (LNNs), we parameterize the kinetic and potential energies with neural networks. The unknown external forces in the Lagrangian formulation are modeled as white Gaussian noise. The corresponding Euler--Lagrange equations then yield a continuous-time stochastic state-space model (SSM) that describes the system dynamics. The neural network parameters and system states are then jointly learned via a maximum-likelihood method using Gaussian-approximation-based Bayesian filters. The effectiveness of the proposed method is demonstrated on pendulum and Duffing oscillator examples, and its performance is compared with conventional LNNs and with approximate Bayesian filters using known system models.

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