Time-Dependent Vector Fields and the Need for Generalization
The fundamental theorem on flows
of the previous page settled the existence and uniqueness theory for systems of ordinary
differential equations on a smooth manifold in the autonomous case. That is, the law specifying
the velocity at each point depends only on the point and not on the moment of time at which the
trajectory passes through it. The autonomous restriction is natural for the geometric content
of the theory (a
vector field
on \(M\) assigns a single vector to each point), and it covers most of the constructions
encountered so far. There are, however, several settings in which an explicit time-dependence
in the velocity law cannot be absorbed into a single vector field on \(M\). They include
classical mechanics with externally driven forces, control systems with prescribed time-varying
inputs, and, more recently, flow-based generative models in machine learning. In the last of
these, a neural network produces a time-dependent velocity field whose flow transports a base
probability distribution into a target one. The present page establishes the existence and
uniqueness theory in this non-autonomous regime, and identifies the time-dependent flow that
takes the role played by the autonomous flow in the previous theorem.
Time-Dependent Vector Fields
Definition: Time-Dependent Vector Field
Let \(M\) be a smooth manifold and \(J \subseteq \mathbb{R}\) an open interval.
A time-dependent vector field on \(M\) with parameter interval
\(J\) is a smooth map
\[
V : J \times M \to TM
\]
such that \(V(t, p) \in T_p M\) for every \((t, p) \in J \times M\). For each
fixed \(t \in J\), the assignment \(V_t : p \mapsto V(t, p)\) is then an
ordinary smooth vector field on \(M\).
The map \(V_t\) is a vector field on \(M\), a slice through the time-dependent
field at time \(t\). The family \(\{ V_t : t \in J \}\) is the data that
replaces a single vector field when the velocity law is allowed to vary with time.
Every ordinary smooth vector field \(X \in \mathfrak{X}(M)\) determines a
time-dependent vector field on \(M\) with parameter interval \(\mathbb{R}\),
through the constant assignment \(V(t, p) = X_p\). The autonomous theory is
therefore contained in the non-autonomous theory as the special case in which
\(V_t\) does not depend on \(t\).
Integral Curves of a Time-Dependent Vector Field
The differential equation that an integral curve of a time-dependent vector field
must satisfy is the natural one. At each parameter value \(t\), the curve must have
velocity equal to the value of the time-dependent vector field at the pair
\((t, \gamma(t))\), evaluated both at the current time and at the current
point.
Definition: Integral Curve of a Time-Dependent Vector Field
Let \(V : J \times M \to TM\) be a time-dependent vector field on \(M\). A
differentiable curve \(\gamma : J_0 \to M\), defined on an open subinterval
\(J_0 \subseteq J\), is an integral curve of \(V\) if
\[
\gamma'(t) = V\bigl(t, \, \gamma(t)\bigr)
\quad \text{for every } t \in J_0 .
\]
When \(V\) is the time-dependent extension of an autonomous vector field
\(X\), that is, when \(V(t, p) = X_p\), the equation
\(\gamma'(t) = V(t, \gamma(t)) = X_{\gamma(t)}\) reduces to the definition of an
integral curve
of \(X\). The non-autonomous definition is therefore a genuine generalization of the
autonomous one, recovering the latter as a special case.
Why a Time-Dependent Vector Field Need Not Generate a Flow
The autonomous fundamental theorem produced not merely an integral curve through each point,
but a smooth flow whose trajectories were the integral curves and whose composition was
governed by the group law of \(\mathbb{R}\). The flow assembled the family of integral curves
into a single map \(\theta : \mathcal{D} \to M\) with the property that any two of its
trajectories through a common point differ only by a shift of time. In the time-dependent case
this consolidation fails at a basic level. Two integral curves of \(V\) that pass through the
same point \(p\) at different parameter values can leave \(p\) with different velocities,
because the velocity of each curve at \(p\) is \(V(t, p)\), where \(t\) is the parameter value
at which that curve passes through \(p\).
Concretely, suppose \(\gamma_1\) and \(\gamma_2\) are two integral curves of \(V\), each with
its own parameter interval, and suppose both pass through the same point \(p\) but at different
parameter values: \(\gamma_1(t_0) = \gamma_2(s_0) = p\) for some \(t_0 \neq s_0\). The
integral-curve equation forces their velocities at \(p\) to be the values of \(V\) at the
corresponding parameter values, namely \(\gamma_1'(t_0) = V(t_0, p)\) and
\(\gamma_2'(s_0) = V(s_0, p)\). These two vectors need not agree. In the autonomous case this
cannot happen, because the velocity at \(p\) is a single vector \(X_p\) independent of any time
parameter. In the time-dependent case the same point \(p\) carries a whole one-parameter family
of possible departure velocities.
The object that replaces the autonomous flow in this setting is therefore not a
map parametrized by a single time variable \(t\), but a map parametrized by a pair
of time variables \((t, t_0)\), the current time and the starting time, together
with the starting point. The next two sections produce this object as a
well-defined smooth map and establish its basic properties. The construction
proceeds by a single technical reduction that transports the entire non-autonomous
theory back to the autonomous theorem already proved.
The Lifting Trick
The reduction from time-dependent to autonomous is geometric in nature. We enlarge
the manifold by one dimension to include the time variable as an additional
coordinate. On the enlarged manifold we construct an honest autonomous vector
field whose flow encodes both the passage of time and the motion produced by the
time-dependent field on the original manifold. This is the lifting trick. The
non-autonomous problem on \(M\) is lifted to an autonomous problem on
\(J \times M\), and the answer on \(J \times M\) is then projected back down to a
time-dependent flow on \(M\). Here and in the definitions above, \(M\) is a smooth manifold without
boundary, so that \(J \times M\) is one as well, as the autonomous theorem applied to it requires.
Construction of the Lifted Vector Field
Let \(V : J \times M \to TM\) be a time-dependent vector field, and let
\(s\) denote the standard coordinate on the open interval \(J \subseteq \mathbb{R}\).
The
product manifold tangent space identification
decomposes the tangent space at each point of \(J \times M\) as a direct sum,
\[
T_{(s, p)} (J \times M) \cong T_s J \oplus T_p M ,
\]
so a tangent vector at \((s, p)\) is specified by giving a tangent vector to \(J\)
at \(s\) and a tangent vector to \(M\) at \(p\). Define
\(\tilde V \in \mathfrak{X}(J \times M)\) by assigning to each point \((s, p)\) the
tangent vector
\[
\tilde V_{(s, p)}
= \left( \frac{\partial}{\partial s}\bigg|_s , \, V(s, p) \right)
\in T_s J \oplus T_p M .
\]
The first component selects the unit tangent vector to \(J\), and the second
component is the value of the time-dependent field at the pair \((s, p)\).
Smoothness of \(V\) as a map \(J \times M \to TM\) implies smoothness of
\(\tilde V\) as a section of \(T(J \times M)\), so \(\tilde V\) is a genuine smooth
autonomous vector field on \(J \times M\).
The Flow of the Lifted Field
Applying the
fundamental theorem on flows
to the autonomous field \(\tilde V\) on the manifold \(J \times M\) produces an
open subset \(\tilde{\mathcal{D}} \subseteq \mathbb{R} \times (J \times M)\) and a
smooth flow
\[
\tilde \theta : \tilde{\mathcal{D}} \to J \times M ,
\]
with the usual properties: \(\tilde \theta(0, (s, p)) = (s, p)\) for every initial
condition, and \(t \mapsto \tilde \theta(t, (s, p))\) is the maximal integral
curve of \(\tilde V\) starting at \((s, p)\). Write the components of
\(\tilde \theta\) explicitly as
\[
\tilde \theta\bigl(t, \, (s, p)\bigr)
= \bigl( \alpha(t, (s, p)), \, \beta(t, (s, p)) \bigr) ,
\]
where \(\alpha : \tilde{\mathcal{D}} \to J\) and
\(\beta : \tilde{\mathcal{D}} \to M\) are smooth.
The curve \(t \mapsto \tilde \theta(t, (s, p))\) is an integral curve of
\(\tilde V\), so its velocity at parameter \(t\) is
\(\tilde V_{\tilde \theta(t, (s, p))}\). Reading off the two components of that
equation through the definition of \(\tilde V\) yields a coupled system for
\(\alpha\) and \(\beta\):
\[
\begin{align*}
\frac{\partial \alpha}{\partial t}(t, (s, p)) &= 1 , \quad \alpha\bigl(0, (s, p)\bigr) = s , \\\\
\frac{\partial \beta}{\partial t}(t, (s, p)) &= V\bigl( \alpha(t, (s, p)), \, \beta(t, (s, p)) \bigr) , \quad \beta\bigl(0, (s, p)\bigr) = p .
\end{align*}
\]
The equation for \(\alpha\) is decoupled and integrates immediately to give
\(\alpha(t, (s, p)) = t + s\). Substituting this into the equation for \(\beta\)
yields
\[
\frac{\partial \beta}{\partial t}(t, (s, p))
= V\bigl( t + s, \, \beta(t, (s, p)) \bigr) .
\]
The reading of the result is geometric. The first component
\(\alpha(t, (s, p)) = t + s\) records that the time coordinate of an integral curve
of \(\tilde V\) advances at unit rate, so that the parameter \(t\) on the integral
curve is identified directly with the elapsed time. The second component \(\beta\)
is a curve in \(M\) whose velocity at parameter \(t\) is the value of the original
time-dependent field at the lifted time \(t + s\) and the current point
\(\beta(t, (s, p))\). When the curve is read in the current time \(t + s\) rather than the elapsed
time \(t\), this is exactly the integral-curve condition for \(V\), starting from \(p\) at time
\(s\).
The Time-Dependent Flow Domain
The starting time of the integral curve in the lifted picture is the first
coordinate \(s\) of the initial condition \((s, p)\). We rename this coordinate
\(t_0\), the starting time of the time-dependent integral curve in the original
manifold \(M\). Writing the elapsed-time parameter \(t\) of the lifted flow as
a current-time parameter via the relation \(t_{\text{current}} = t_0 + t\), we
replace the lifted parameters \((t, (t_0, p)) \in \tilde{\mathcal{D}}\) by the
parameter triple \((t_{\text{current}}, t_0, p)\). The subset of
\(\mathbb{R} \times J \times M\) on which this reparametrization makes sense is
\[
\mathcal{E}
= \bigl\{ (t, t_0, p) \in \mathbb{R} \times J \times M
: (t - t_0, \, (t_0, p)) \in \tilde{\mathcal{D}} \bigr\} .
\]
Openness of \(\tilde{\mathcal{D}}\) in
\(\mathbb{R} \times (J \times M)\) and continuity of the map
\((t, t_0, p) \mapsto (t - t_0, (t_0, p))\) imply openness of \(\mathcal{E}\) in
\(\mathbb{R} \times J \times M\). Moreover, the membership
\((t - t_0, (t_0, p)) \in \tilde{\mathcal{D}} \subseteq \mathbb{R} \times
(J \times M)\) forces the initial-condition component to lie in \(J \times M\),
so \(t_0 \in J\) automatically. The identity \(\alpha(t - t_0, (t_0, p)) =
(t - t_0) + t_0 = t\) shows that whenever \((t, t_0, p) \in \mathcal{E}\) the
current time \(t\) lies in \(J\) as well, since \(\alpha\) takes values in
\(J\). Hence
\[
\mathcal{E} \subseteq J \times J \times M ,
\]
and the time-dependent flow we are about to define will take \((t, t_0, p)\) with
\(t, t_0 \in J\). The construction of the flow itself, together with the statement
and proof of its defining properties, is the subject of the next section.
The Fundamental Theorem on Time-Dependent Flows
The lifted construction produces, by projection onto the second factor, the map
that plays the role of the flow in the time-dependent regime. Where the autonomous
flow is parametrized by a single time variable, the time-dependent flow is
parametrized by current time, starting time, and starting point. The position reached from \(p\)
can depend on the starting time \(t_0\) itself, not only on the elapsed time \(t - t_0\).
Definition: Time-Dependent Flow
Let \(V : J \times M \to TM\) be a smooth time-dependent vector field, and let
\(\tilde V\) be the lifted vector field on \(J \times M\) with flow
\(\tilde \theta = (\alpha, \beta) : \tilde{\mathcal{D}} \to J \times M\)
constructed in the previous section. The time-dependent flow of
\(V\) is the map
\[
\psi : \mathcal{E} \to M , \quad
\psi(t, t_0, p) = \beta\bigl( t - t_0, \, (t_0, p) \bigr) ,
\]
where \(\mathcal{E} \subseteq J \times J \times M\) is the open subset defined
in the lifting construction.
Smoothness of \(\beta\) on \(\tilde{\mathcal{D}}\), smoothness of the map
\((t, t_0, p) \mapsto (t - t_0, (t_0, p))\), and openness of \(\mathcal{E}\) imply smoothness
of \(\psi\) on \(\mathcal{E}\). The differential equation
satisfied by \(\beta\) translates directly into an equation for \(\psi\). For fixed
starting data \((t_0, p)\) and varying current time \(t\), set
\(\psi^{(t_0, p)}(t) = \psi(t, t_0, p) = \beta(t - t_0, (t_0, p))\). After the
renaming \(s = t_0\) of the lifting construction, the equation derived in the
previous section reads
\(\partial \beta / \partial \tau (\tau, (t_0, p)) = V(\tau + t_0, \beta(\tau,
(t_0, p)))\). The composition \(t \mapsto \beta(t - t_0, (t_0, p))\) has inner
derivative \(d(t - t_0)/dt = 1\). Applying the chain rule and substituting
\(\tau = t - t_0\) gives
\[
\begin{align*}
\bigl( \psi^{(t_0, p)} \bigr)'(t)
&= \frac{\partial \beta}{\partial \tau}(t - t_0, (t_0, p)) \\\\
&= V\bigl( (t - t_0) + t_0, \, \beta(t - t_0, (t_0, p)) \bigr) \\\\
&= V\bigl( t, \, \psi^{(t_0, p)}(t) \bigr) ,
\end{align*}
\]
with initial value \(\psi^{(t_0, p)}(t_0) = \beta(0, (t_0, p)) = p\), so
\(\psi^{(t_0, p)}\) is an integral curve of \(V\) with starting time \(t_0\) and
starting point \(p\). The defining properties of \(\psi\) are collected in the
theorem below.
The Fundamental Theorem
The autonomous fundamental theorem produced a maximal flow with three structural
properties: smoothness, the group law, and characterization of each trajectory as
a maximal integral curve. The time-dependent theorem records the corresponding
structural data, with the group law replaced by a two-parameter cocycle relation
that reflects the non-autonomous nature of the trajectories.
Theorem (Fundamental Theorem on Time-Dependent Flows)
Let \(M\) be a smooth manifold without boundary, let \(J \subseteq \mathbb{R}\) be an open
interval, and let \(V : J \times M \to TM\) be a smooth time-dependent vector
field on \(M\). There exist an open subset
\(\mathcal{E} \subseteq J \times J \times M\) and a smooth map
\(\psi : \mathcal{E} \to M\), the time-dependent flow of \(V\), with the
following properties.
(a) Existence and uniqueness of trajectories.
For every
\(t_0 \in J\) and \(p \in M\), the set
\(\mathcal{E}^{(t_0, p)} = \{ t \in J : (t, t_0, p) \in \mathcal{E} \}\) is
an open interval containing \(t_0\). The smooth curve
\(\psi^{(t_0, p)} : \mathcal{E}^{(t_0, p)} \to M\) defined by
\(\psi^{(t_0, p)}(t) = \psi(t, t_0, p)\) is the unique maximal integral
curve of \(V\) satisfying \(\psi^{(t_0, p)}(t_0) = p\).
(b) Reparametrization invariance.
If
\(t_1 \in \mathcal{E}^{(t_0, p)}\) and \(q = \psi^{(t_0, p)}(t_1)\), then
\(\mathcal{E}^{(t_1, q)} = \mathcal{E}^{(t_0, p)}\) and
\(\psi^{(t_1, q)} = \psi^{(t_0, p)}\).
(c) Time-slice diffeomorphisms.
For each \((t_1, t_0) \in
J \times J\), the set
\(M_{t_1, t_0} = \{ p \in M : (t_1, t_0, p) \in \mathcal{E} \}\) is open in
\(M\), and the map \(\psi_{t_1, t_0} : M_{t_1, t_0} \to M\) defined by
\(\psi_{t_1, t_0}(p) = \psi(t_1, t_0, p)\) is a diffeomorphism onto
\(M_{t_0, t_1}\), with inverse \(\psi_{t_0, t_1}\).
(d) Cocycle relation.
Whenever \(p \in M_{t_1, t_0}\) and
\(\psi_{t_1, t_0}(p) \in M_{t_2, t_1}\), we have \(p \in M_{t_2, t_0}\) and
\[
\psi_{t_2, t_1} \circ \psi_{t_1, t_0}(p)
= \psi_{t_2, t_0}(p) .
\]
The cocycle relation in (d) is the two-parameter replacement for the group law
\(\theta_{t_2} \circ \theta_{t_1} = \theta_{t_1 + t_2}\) of an autonomous flow. The
autonomous flow can be recovered from the time-dependent flow when \(V\) is
constant in \(t\). The time-slice diffeomorphism \(\psi_{t_1, t_0}\) then depends
only on the difference \(t_1 - t_0\), and the cocycle relation collapses to the
additive group law in that difference.
Proof (outline):
Every assertion is read off from the corresponding property of the lifted flow
\(\tilde \theta = (\alpha, \beta)\) on \(J \times M\), which exists and is
smooth by the autonomous
fundamental theorem on flows
applied to the lifted field.
(a).
The map \(t \mapsto \psi^{(t_0, p)}(t)\) is an integral
curve of \(V\) with \(\psi^{(t_0, p)}(t_0) = p\) by the differential equation
derived above. For uniqueness and maximality, suppose
\(\gamma : J_0 \to M\) is any integral curve of \(V\) with
\(\gamma(t_0) = p\), and define a lifted curve
\[
\tilde \gamma : J_0 \to J \times M
\]
by \(\tilde \gamma(t) = (t, \gamma(t))\). The velocity of \(\tilde \gamma\) at \(t\) is
\[
\begin{align*}
\bigl(\partial / \partial s |_t, \, \gamma'(t)\bigr)
&= \bigl(\partial / \partial s |_t, \, V(t, \gamma(t))\bigr) \\\\
&= \tilde V_{\tilde \gamma(t)}.
\end{align*}
\]
Thus, \(\tilde \gamma\) is an integral curve of \(\tilde V\) with \(\tilde \gamma(t_0) = (t_0, p)\).
The shifted curve
\[
t \mapsto \tilde \theta(t - t_0, (t_0, p))
\]
is also an integral curve of \(\tilde V\) by the
translation lemma,
and it is defined exactly for those \(t\) with \(t - t_0\) in the lifted maximal interval
\(\tilde{\mathcal{D}}^{(t_0, p)} = \{ \tau \in \mathbb{R} : (\tau, (t_0, p)) \in \tilde{\mathcal{D}} \}\).
It too takes the value \((t_0, p)\) at
\(t = t_0\), since \(\tilde \theta(0, (t_0, p)) = (t_0, p)\).
Both curves therefore meet \((t_0, p)\) at time \(t_0\), and we compare them by
shifting that time to the origin.
Applied to \(\tilde \gamma\), the translation lemma gives the integral curve
\(t \mapsto \tilde \gamma(t + t_0)\) on \(J_0 - t_0\), whose starting point is
\((t_0, p)\). The
absorption of integral curves by trajectories
therefore gives \(J_0 - t_0 \subseteq \tilde{\mathcal{D}}^{(t_0, p)}\) and
identifies that translate with the restriction of
\(t \mapsto \tilde \theta(t, (t_0, p))\) to \(J_0 - t_0\). Translating back
identifies \(\tilde \gamma\) as the restriction of the shifted curve to
\(J_0\), with
\[
J_0 \subseteq t_0 + \tilde{\mathcal{D}}^{(t_0, p)} = \mathcal{E}^{(t_0, p)}.
\]
Projecting to the \(M\)-component gives \(\gamma\) as the restriction
of \(\psi^{(t_0, p)}\) to \(J_0\). The set \(\mathcal{E}^{(t_0, p)}\) is
therefore the maximal interval on which the integral curve through
\((t_0, p)\) exists, and is an open interval containing \(t_0\) since it is
the affine shift of an open interval in the lifted flow domain.
(b).
Suppose \(t_1 \in \mathcal{E}^{(t_0, p)}\) and set
\(q = \psi^{(t_0, p)}(t_1)\). Both \(\psi^{(t_0, p)}\) and \(\psi^{(t_1, q)}\)
are integral curves of \(V\) that pass through \(q\) at time \(t_1\). Part
(a) applied at \((t_1, q)\) exhibits \(\psi^{(t_0, p)}\) as a restriction of
\(\psi^{(t_1, q)}\), so
\(\mathcal{E}^{(t_0, p)} \subseteq \mathcal{E}^{(t_1, q)}\) and the two agree
there. In particular \(t_0\) lies in \(\mathcal{E}^{(t_1, q)}\) and
\(\psi^{(t_1, q)}(t_0) = p\), so the same argument run at \((t_0, p)\) gives
the reverse containment. The domains coincide and the curves are equal.
(c).
Openness of \(M_{t_1, t_0}\) in \(M\) follows from openness of \(\mathcal{E}\). The containment
\(\psi_{t_1, t_0}(M_{t_1, t_0}) \subseteq M_{t_0, t_1}\) is part (b) read as a
statement on domains. For \(p \in M_{t_1, t_0}\) and
\(q = \psi_{t_1, t_0}(p)\), part (b) gives
\[
\mathcal{E}^{(t_1, q)} = \mathcal{E}^{(t_0, p)},
\]
so \(t_0 \in \mathcal{E}^{(t_1, q)}\), which is exactly
\(q \in M_{t_0, t_1}\). Reversing the roles of \(t_0\) and \(t_1\) gives the
opposite containment
\(\psi_{t_0, t_1}(M_{t_0, t_1}) \subseteq M_{t_1, t_0}\), and the cocycle
relation (d) applied with \(t_2 = t_0\) yields
\[
\psi_{t_0, t_1} \circ \psi_{t_1, t_0} = \psi_{t_0, t_0} = \mathrm{id}
\]
on \(M_{t_1, t_0}\). Here we use the identity
\[
\psi_{t_0, t_0}(p) = \psi^{(t_0, p)}(t_0) = p ,
\]
which holds by (a). Together these identify \(\psi_{t_1, t_0}\) as a smooth bijection
\(M_{t_1, t_0} \to M_{t_0, t_1}\) with smooth inverse \(\psi_{t_0, t_1}\),
which is the definition of a diffeomorphism.
(d).
Suppose \(p \in M_{t_1, t_0}\) and
\(\psi_{t_1, t_0}(p) \in M_{t_2, t_1}\), and set \(q = \psi_{t_1, t_0}(p) =
\psi^{(t_0, p)}(t_1)\). The hypothesis \(q \in M_{t_2, t_1}\) says that
\(t_2 \in \mathcal{E}^{(t_1, q)}\), and part (b) identifies that interval
with \(\mathcal{E}^{(t_0, p)}\). Hence \(t_2 \in \mathcal{E}^{(t_0, p)}\),
which is exactly \(p \in M_{t_2, t_0}\), and both sides of the asserted
identity are defined. Part (b) then gives
\[
\psi^{(t_1, q)}(t_2) = \psi^{(t_0, p)}(t_2),
\]
which unwinds to
\[
\psi_{t_2, t_1} \circ \psi_{t_1, t_0}(p) = \psi_{t_2, t_0}(p) .
\]
The logical dependence among the four parts does not follow the order in which
they are stated. Existence and uniqueness in (a) is the input that makes the
reparametrization identity in (b) available. The cocycle relation in (d) is a
corollary of (b) read at the level of time-slice maps, and the diffeomorphism
property in (c) is then a consequence of (b) and (d) combined. The interlocking
of the four properties is exactly what makes the time-dependent flow a coherent
geometric object rather than a loose collection of integral curves.
Flow Matching and the Time-Dependent Flow
A modern application in which the time-dependent flow plays an essential role lies outside
classical mechanics or control theory altogether: in the construction of generative models in
machine learning. The strategy is to learn a time-dependent velocity field whose flow
transports a base probability distribution onto a target one, and to sample from the target by
integrating the learned field from a sample of the base distribution. The target may be the
distribution of natural images, of natural-language text, or of any other complex data type.
The theorem of the previous section is a rigorous foundation on which this construction rests.
When the learned velocity field is smooth, the theorem guarantees that this field generates a
well-defined two-parameter family of diffeomorphisms \(\psi_{t_1, t_0}\) between open subsets,
each of which can be evaluated by ODE integration, and that these diffeomorphisms satisfy the
cocycle relation, which ensures consistency when these integrations are composed.
The Time-Dependent Flow in Generative Modeling
In the flow-matching construction
for generative models, a neural network is trained to approximate a time-dependent
velocity field \(\mathbf{u}_t(\mathbf{x})\) on \(\mathbb{R}^n\). That field is
specified only on the closed unit interval and can be singular at an endpoint, but a
network with smooth activations and a smooth time input defines a smooth field for
all times. The learned field is a time-dependent vector field in the sense of this
page, with a parameter interval \(J\) that is an open interval containing the closed
unit interval, as the construction has to evaluate the flow at both endpoints \(0\)
and \(1\). The generative procedure is the time-slice diffeomorphism \(\psi_{1, 0}\)
of the corresponding time-dependent flow, defined on the open set \(M_{1, 0}\) of
starting points whose trajectory survives to time \(1\). A sample \(\mathbf{x}_0\)
drawn from the base distribution at time \(0\) is carried by the flow to
\(\psi_{1, 0}(\mathbf{x}_0)\). Provided the base distribution is concentrated on
\(M_{1, 0}\), the distribution of \(\psi_{1, 0}(\mathbf{x}_0)\) is the pushforward
of the base distribution under \(\psi_{1, 0}\). This pushforward approximates the
target distribution to the extent that the learned field reproduces the intended
family of distributions from the base at time \(0\) to the target at time \(1\). The
well-posedness of the generative procedure was previously available through the
Lipschitz
version of the Picard-Lindelöf theorem on Euclidean space, which gives
existence and uniqueness of integral curves for each fixed sample but treats each
trajectory in isolation.
The fundamental theorem of this page does more. It produces \(\psi_{1, 0}\) as a single
smooth diffeomorphism between open subsets of \(\mathbb{R}^n\), with composition behavior
governed by the cocycle relation. The diffeomorphism property underlies the
change-of-variables formula for probability densities under the flow, and the cocycle
relation underlies the consistency of partial integrations. Combined with the standard
smooth dependence of solutions to ODEs on parameters, the smoothness of \(\psi\) jointly in
current time, starting time, and starting point underlies the differentiability of the
generative procedure with respect to the parameters of the learned field. Backpropagation
through the flow during training exploits exactly that differentiability.
The connection runs in both directions, and the geometric content of the time-dependent flow
has a corresponding payoff outside Euclidean settings. Data are sometimes naturally modeled as
a manifold rather than as \(\mathbb{R}^n\). Rotations, points on a sphere, and strictly
positive probability distributions over a finite set are all of this kind. For such data the
Euclidean Picard-Lindelöf statement no longer applies directly, but the manifold-level
fundamental theorem of this page does. The flow-generation step of flow matching therefore
transfers to manifold-valued data, once paths of distributions and the velocity fields that
transport them have been defined on the manifold. Defining them in practice uses additional
structure, such as a Riemannian metric supplying the interpolating curves and the norm in the
training loss. The time-dependent flow then produces a diffeomorphism between open subsets of
the data manifold by exactly the same mechanism. The technical apparatus developed here is, in
this sense, the manifold-level foundation that flow-based generative modeling needs in order to
make the leap from Euclidean to geometric data.