A FreeFM tutorial with damped oscillators and Lorenz-63

TL;DR. FreeFM is a training-free, memory-based flow matching forecaster. It builds a bank of observed transition pairs, uses those pairs to define a closed-form empirical velocity field, and produces probabilistic forecasts by integrating that field. On smooth systems such as the damped harmonic oscillator, it works very well when the memory bank covers the relevant phase space region. On chaotic systems such as Lorenz-63, short-term forecasts can be meaningful, but long-horizon pointwise matching remains fragile.

Why FreeFM?

Most flow matching models learn a neural velocity field. FreeFM asks a simpler question:

Can we build a probabilistic forecaster directly from observed transition pairs, without training a neural network?

The answer is yes, provided we understand the limitation: FreeFM does not extrapolate like a learned global model. It retrieves and blends local transition geometry from the data.

This makes FreeFM useful as both:

1. a lightweight forecasting baseline, and
2. a diagnostic lens for understanding what finite-sample flow matching is doing.

Method in one picture

Given transition pairs $(x_0^{(j)},x_1^{(j)})$ extracted from a time series dataset, FreeFM defines Gaussian interpolation paths with mean and variance:

$$m_t^{(j)} = (1-t)x_0^{(j)} + t x_1^{(j)}, \qquad c_t^2 = \sigma_{\min}^2 + \sigma^2 t(1-t).$$

The empirical flow matching velocity is

$$v(t,z) = G_t z + \sum_j \alpha_j(t,z) ( (x_1^{(j)}-x_0^{(j)}) - G_t m_t^{(j)} ), \qquad G_t = \frac{\sigma^2(1-2t)}{2c_t^2},$$

with Gaussian responsibilities

$$ \alpha_j(t,z) \propto \exp\left( -\frac{|z-m_t^{(j)}|^2}{2c_t^2} \right). $$

A one-step forecast samples $Z_0 \sim \mathcal{N}(x_k,\sigma_{\min}^2I)$, then integrates $\frac{dZ_t}{dt}=v(t,Z_t),$ for $t \in [0,1]$, and returns $Z_1$ as a predictive sample for $x_{k+1}$. Repeating this autoregressively gives a probabilistic trajectory forecast.

Two memory bank regimes

The same FreeFM algorithm can answer two different forecasting questions depending on how the transition memory bank is constructed.

This distinction is central. FreeFM is only as good as the local transition coverage in its memory bank.

Results summary

Part I — Many-trajectory/source-style setup

In the source-style setup, we simulate multiple independent trajectories and split them by trajectory:

all_seqs = make_dataset(...)
train_val_seqs = all_seqs[:-N_test]
test_seqs = all_seqs[-N_test:]
train_seqs, val_seqs = split_train_val(train_val_seqs)

The memory bank pools transitions across all training trajectories:

X0_train, X1_train = build_memory_bank_np(train_seqs)

This asks whether local transition geometry from many example trajectories can forecast a new trajectory.

Demo A: damped harmonic oscillator

The damped harmonic oscillator is a good sanity check: the state is two-dimensional, the phase portrait is a smooth spiral, and nearby transitions are informative.

The validation search selects $\sigma_{\min}=0.01$, $\sigma=0.01.$

The held-out forecast is accurate both coordinate-wise and in phase space.

Takeaway. When the memory bank contains nearby spirals, FreeFM can reuse local transition directions almost exactly.

Demo B: Lorenz-63

Lorenz-63 is a stress test because it is chaotic. Long-horizon pointwise prediction is not expected to stay synchronized indefinitely.

The validation search selects $\sigma_{\min}=0.02$, $\sigma=0.01.$

The short-window MSE over the first 50 forecast steps is $5.83\times 10^{-3}$, while the full-horizon MSE increases to $1.82\times 10^{-1}$.

Takeaway. FreeFM can track local dynamics, but chaotic systems punish small autoregressive errors. Statistical and phase-space diagnostics are more meaningful than full-horizon pointwise MSE alone.

Part II — Single-trajectory/time-series setup

The single-trajectory setup is closer to classical forecasting. We observe one ordered path $X_0,X_1,\ldots,X_{T_{\mathrm{obs}}-1},$ and forecast the held-out future $X_{T_{\mathrm{obs}}},\ldots,X_{T_{\mathrm{obs}}+H-1}.$

The observed prefix is split into training and validation segments:

training segment | validation segment | held-out future

During validation, FreeFM uses only the training segment. After choosing $(\sigma_{\min},\sigma)$, we rebuild the final memory bank using all observed transitions before the forecast boundary.

Single-trajectory DHO

In this regime, the memory bank comes from one observed damped spiral rather than from many independent trajectories. This is more fragile because the future may leave the observed local support.

The selected hyperparameters are $\sigma_{\min}=0.03$, $\sigma=0.01.$

The final memory bank contains 399 train-plus-validation transitions. The forecast MSE is $1.04\times 10^{-2}$, with terminal error $0.309$.

Single-trajectory Lorenz-63

The selected hyperparameters are $\sigma_{\min}=0.01$, $\sigma=0.01.$

The final memory bank contains 399 train-plus-validation transitions. The forecast MSE is $1.27\times 10^{-2}$, with terminal error $0.360$.

Practical notes

• Use topk=None for the exact dense empirical field on small memory banks.
• Use finite topk for faster nearest-neighbor approximation on larger memory banks.
• Larger topk, more ODE steps, and RK4 can give smoother but slower forecasts.
• For Lorenz-63, exact long-horizon path matching is not expected because the system is chaotic.
• For noisy dynamical systems, inject fresh uncertainty at each autoregressive step.

Main takeaway

FreeFM is best understood as a memory-based flow matching forecaster. It replaces parametric learning with local transition retrieval.

Its strengths are:

• no neural network training,
• simple implementation,
• interpretable local transition reuse,
• natural probabilistic forecasts through ensembles.

Its weaknesses are:

• dependence on memory bank coverage,
• nearest-neighbor cost for large datasets,
• limited extrapolation outside observed support,
• long-horizon fragility on chaotic systems.

On the one hand, FreeFM can forecast without training, but it cannot forecast accurately without sufficient coverage. This opens up an interesting direction: using FreeFM as a methodological bridge between nonparametric memory-based forecasting and learned flow matching models for sequential data.

On the other hand, while the above two regimes of many independent trajectories vs. one long trajectory is useful pedagogically, real forecasting data often sit between these two extremes. Developing principled methods that work well in the intermediate regimes is an important direction for future work.