SlowGuess/NeurIPS_2023_OCR huggingface dataset:
$$
\begin{array}{l} \leq \left| \frac {\partial p (x , t)}{\partial t} - \frac {\partial \tilde {p} (x , t)}{\partial t} \right| + \sum_ {i = 1} ^ {d} \left| \mu_ {i} (x, t) \frac {\partial p (x , t)}{\partial x _ {i}} - \mu_ {i} (x, t) \frac {\partial \tilde {p} (x , t)}{\partial x _ {i}} \right| + \sum_ {i = 1} ^ {d} \left| \frac {\partial \mu_ {i} (x , t)}{\partial x _ {i}} p (x, t) - \frac {\partial \mu_ {i} (x , t)}{\partial x _ {i}} \tilde {p} (x, t) \right| \\ + \left| \sum_ {i = 1} ^ {d} \sum_ {j = 1} ^ {d} D _ {i j} \frac {\partial^ {2}}{\partial x _ {i} \partial x _ {j}} p (x, t) - D _ {i j} \frac {\partial^ {2}}{\partial x _ {i} \partial x _ {j}} \tilde {p} (x, t) \right| \\ \leq \left| \frac {\partial p (x , t)}{\partial t} - \frac {\partial \tilde {p} (x , t)}{\partial t} \right| + R _ {\mu} \sum_ {i = 1} ^ {d} \left| \frac {\partial p (x , t)}{\partial x _ {i}} - \frac {\partial \tilde {p} (x , t)}{\partial x _ {i}} \right| + d R _ {\mu_ {p}} (p (x, t) - \tilde {p} (x, t)) \\ + R _ {d} \sum_ {i = 1} ^ {d} \sum_ {j = 1} ^ {d} \left| \frac {\partial^ {2}}{\partial x _ {i} \partial x _ {j}} p (x, t) - \frac {\partial^ {2}}{\partial x _ {i} \partial x _ {j}} \tilde {p} (x, t) \right| \\ = 2 \left| \frac {\partial \psi (x , t)}{\partial t} ^ {\top} (M - \tilde {P} M \tilde {P}) \psi (x, t) \right| + 2 R _ {\mu} \sum_ {i = 1} ^ {d} \left| \frac {\partial \psi (x , t)}{\partial x _ {i}} ^ {\top} (M - \tilde {P} M \tilde {P}) \psi (x, t) \right| + d R _ {\mu_ {p}} | \psi (x, t) | ^ {\top} (M - \tilde {P} M \tilde {P}) \psi (x, t) | \\ + 2 R _ {d} \left| \psi_ {j j} ^ {\prime \prime} (x, t) ^ {\top} \left(M - \tilde {P} M \tilde {P}\right) \psi (x, t) + \psi_ {i} ^ {\prime} (x, t) ^ {\top} \left(M - \tilde {P} M \tilde {P}\right) \psi_ {j} ^ {\prime} (x, t) \right| \\ \leq 2 \left| \frac {\partial \psi (x , t)}{\partial t} ^ {\top} (M - \tilde {P} M \tilde {P}) \psi (x, t) \right| + 2 R _ {\mu} \sum_ {i = 1} ^ {d} \left| \frac {\partial \psi (x , t)}{\partial x _ {i}} ^ {\top} (M - \tilde {P} M \tilde {P}) \psi (x, t) \right| + d R _ {\mu_ {p}} | \psi (x, t) ^ {\top} (M - \tilde {P} M \tilde {P}) \psi (x, t) | \\ + 2 R _ {d} \left| \psi_ {j i} ^ {\prime \prime} (x, t) ^ {\top} \left(M - \tilde {P} M \tilde {P}\right) \psi (x, t) \right| + 2 R _ {d} \left| \psi_ {i} ^ {\prime} (x, t) ^ {\top} \left(M - \tilde {P} M \tilde {P}\right) \psi_ {j} ^ {\prime} (x, t) \right|. \\ \end{array}
$$
Now, combining everything together, we get
$$
\begin{array}{l} \left| \frac {\partial p (x , t)}{\partial t} - \frac {\partial \tilde {p} (x , t)}{\partial t} + \sum_ {i = i} ^ {d} \frac {\partial (\mu_ {i} (x , t) p (x , t))}{\partial x _ {i}} - \frac {\partial (\mu_ {i} (x , t) \tilde {p} (x , t))}{\partial x _ {i}} - \sum_ {i = 1} ^ {d} \sum_ {j = 1} ^ {d} D _ {i j} \frac {\partial^ {2}}{\partial x _ {i} \partial x _ {j}} p (x, t) - D _ {i j} \frac {\partial^ {2}}{\partial x _ {i} \partial x _ {j}} \tilde {p} (x, t) \right| \\ \leq 2 c _ {M} \tilde {u} _ {1} (x) \tilde {u} _ {t} (x) + 2 c _ {M} ^ {1 / 2} f (x; M, \psi) ^ {1 / 2} \tilde {u} _ {t} (x) + 2 c _ {M} ^ {1 / 2} \tilde {f} _ {t} (x; M, \psi) ^ {1 / 2} \tilde {u} _ {1} (x) + d R _ {\mu_ {p}} (2 c _ {M} ^ {1 / 2} f (x; M, \psi) ^ {1 / 2} \tilde {u} _ {1} (x) + c _ {M} \tilde {u} _ {1} (x) ^ {2}) \\ + 2 R _ {\mu} \sum_ {i = 1} ^ {d} \left[ c _ {M} \tilde {u} _ {1} (x) \tilde {u} _ {x _ {i}} (x) + c _ {M} ^ {1 / 2} f (x; M, \psi) ^ {1 / 2} \tilde {u} _ {x _ {i}} (x) + c _ {M} ^ {1 / 2} \tilde {f} _ {x _ {i}} (x; M, \psi) ^ {1 / 2} \tilde {u} _ {1} (x) \right] \\ + 2 R _ {d} \sum_ {i = 1} ^ {d} \sum_ {j = 1} ^ {d} \left[ c _ {M} \tilde {u} _ {1} (x) \tilde {u} _ {x _ {j} x _ {i}} (x) + c _ {M} ^ {1 / 2} f (x; M, \psi) ^ {1 / 2} \tilde {u} _ {x _ {j} x _ {i}} (x) + c _ {M} ^ {1 / 2} \tilde {f} _ {x _ {j} x _ {i}} (x; M, \psi) ^ {1 / 2} \tilde {u} _ {1} (x) \right] \\ + 2 R _ {d} \sum_ {i = 1} ^ {d} \sum_ {j = 1} ^ {d} \left[ c _ {M} \tilde {u} _ {x _ {j}} (x) \tilde {u} _ {x _ {i}} (x) + c _ {M} ^ {1 / 2} f _ {x _ {i}} (x; M, \psi) ^ {1 / 2} \tilde {u} _ {x _ {j}} (x) + c _ {M} ^ {1 / 2} \tilde {f} _ {x _ {j}} (x; M, \psi) ^ {1 / 2} \tilde {u} _ {x _ {i}} (x) \right]. \\ \end{array}
$$
Now from theorems 6 and 7, we have that when the fill distance  $h < \frac{\sigma}{C'}$  with  $\sigma = \min(R, \frac{1}{\tau})$ , then
$$
\left\| \tilde {u} _ {1} \right\| _ {L ^ {\infty} (\mathcal {X})} \leq C q _ {\eta} e ^ {- \frac {c \sigma}{h} \log \frac {c \sigma}{h}}, \left\| \tilde {u} _ {t} \right\| _ {L ^ {\infty} (\mathcal {X})} \leq C q _ {\eta} e ^ {- \frac {c \sigma}{h} \log \frac {c \sigma}{h}}, \left\| \tilde {u} _ {x _ {i}} \right\| _ {L ^ {\infty} (\mathcal {X})} \leq C q _ {\eta} e ^ {- \frac {c \sigma}{h} \log \frac {c \sigma}{h}} \tag {48}
$$
$$
\text {a n d} \| \tilde {u} _ {x _ {i} x _ {j}} \| _ {L ^ {\infty} (\mathcal {X})} \leq C q _ {\eta} e ^ {- \frac {c \sigma}{h} \log \frac {c \sigma}{h}}
$$
for all  $i \in \{1, \dots, d\}$ ,  $j \in \{1, \dots, d\}$  and some constant  $C, c, C'$  depending only on  $d$ . Hence, we have
$$
\begin{array}{l} \left| \frac {\partial (p (x , t) - \tilde {p} (x , t))}{\partial t} + \sum_ {i = i} ^ {d} \frac {\partial (\mu_ {i} (x , t) (p (x , t) - \tilde {p} (x , t)))}{\partial x _ {i}} - \sum_ {i = 1} ^ {d} \sum_ {j = 1} ^ {d} D _ {i j} \frac {\partial^ {2}}{\partial x _ {i} \partial x _ {j}} (p (x, t) - \tilde {p} (x, t)) \right| \\ \leq \left(2 + d R _ {\mu} + 2 d R _ {\mu} + 4 d ^ {2} R _ {d}\right) \| M \| C ^ {2} q _ {\eta} ^ {2} e ^ {- \frac {2 c \sigma}{h} \log \frac {c \sigma}{h}} + \left(2 + d R _ {\mu_ {p}} + 2 d R _ {\mu} + 2 d ^ {2} R _ {d}\right) \sqrt {\| M \| p (x , t)} C q _ {\eta} e ^ {- \frac {c \sigma}{h} \log \frac {c \sigma}{h}} \\ + 2 C q _ {\eta} e ^ {- \frac {c \sigma}{h} \log \frac {c \sigma}{h}} \sqrt {\| M \| \tilde {f} _ {t} (x ; M , \psi)} + (2 R _ {\mu} + 4 d R _ {d}) C q _ {\eta} e ^ {- \frac {c \sigma}{h} \log \frac {c \sigma}{h}} \sum_ {i = 1} ^ {d} \sqrt {\| M \| \tilde {f} _ {x _ {i}} (x ; M , \psi)} \\ + 2 R _ {d} C q _ {\eta} e ^ {- \frac {c \sigma}{h} \log \frac {c \sigma}{h}} \sum_ {i = 1} ^ {d} \sum_ {j = 1} ^ {d} \sqrt {\| M \| \tilde {f} _ {x _ {j} x _ {i}} (x ; M , \psi)}. \\ \end{array}
$$
A.6 Final Approximation Bound (Proof of Theorem 2)
Proof. From the previous result in theorem 1, we know that there exists an  $\mathsf{M}_{\varepsilon} \in \mathbb{S}_{+}(\mathcal{H}_{X} \otimes \mathcal{H}_{T})$  with  $\mathrm{rank}(\mathsf{M}_{\varepsilon}) \leq q$ , such that for the representation  $\hat{p}(x,t) = \phi_{X,T}^{\top}\mathsf{M}_{\varepsilon}\phi_{X,T}$ , following holds under
assumption 2 on the coefficients of the fractional FPE,
$$
\left\| \frac {\partial \hat {p} (x , t)}{\partial t} + \sum_ {i = 1} ^ {d} \frac {\partial}{\partial x _ {i}} \left(\mu_ {i} (x, t) \hat {p} (x, t)\right) - \sum_ {i = 1} ^ {d} \sum_ {j = 1} ^ {d} D _ {i j} \frac {\partial^ {2}}{\partial x _ {i} \partial x _ {j}} \hat {p} (x, t) \right\| _ {L ^ {2} (\tilde {\mathfrak {X}})} = O (\varepsilon), \tag {49}
$$
for an appropriate choice of parameters. We also know that
$$
\frac {\partial p ^ {\star} (x , t)}{\partial t} + \sum_ {i = 1} ^ {d} \frac {\partial}{\partial x _ {i}} \left(\mu_ {i} (x, t) p ^ {\star} (x, t)\right) - \sum_ {i = 1} ^ {d} \sum_ {j = 1} ^ {d} D _ {i j} \frac {\partial^ {2}}{\partial x _ {i} \partial x _ {j}} p ^ {\star} (x, t) = 0. \tag {50}
$$
We know that, for given  $A_{m}$ ,  $\tilde{p} (x,t) = \psi (x,t)^{\top}\tilde{P}\mathsf{M}_{\varepsilon}\tilde{P}\psi (x,t)^{\top}$  where  $\tilde{P}$  is the projection operator defined earlier. From the results in theorem 8, we have
$$
\begin{array}{l} \left| \frac {\partial (\hat {p} (x , t) - \tilde {p} (x , t))}{\partial t} + \sum_ {i = i} ^ {d} \frac {\partial (\mu_ {i} (x , t) (\hat {p} (x , t) - \tilde {p} (x , t)))}{\partial x _ {i}} - \sum_ {i = 1} ^ {d} \sum_ {j = 1} ^ {d} D _ {i j} \frac {\partial^ {2}}{\partial x _ {i} \partial x _ {j}} (\hat {p} (x, t) - \tilde {p} (x, t)) \right| \\ \leq (2 + d R _ {\mu} + 2 d R _ {\mu} + 4 d ^ {2} R _ {d}) \| \mathsf {M} _ {\varepsilon} \| C ^ {2} q _ {\eta} ^ {2} e ^ {- \frac {2 c \sigma}{h} \log \frac {c \sigma}{h}} + (2 + d R _ {\mu_ {p}} + 2 d R _ {\mu} + 2 d ^ {2} R _ {d}) \sqrt {\| \mathsf {M} _ {\varepsilon} \| p (x , t)} C q _ {\eta} e ^ {- \frac {c \sigma}{h} \log \frac {c \sigma}{h}} \\ + 2 C q _ {\eta} e ^ {- \frac {c \sigma}{h} \log \frac {c \sigma}{h}} \sqrt {\| \mathsf {M} _ {\varepsilon} \| \tilde {f} _ {t} (x ; M , \psi)} + (2 R _ {\mu} + 4 d R _ {d}) C q _ {\eta} e ^ {- \frac {c \sigma}{h} \log \frac {c \sigma}{h}} \sum_ {i = 1} ^ {d} \sqrt {\| \mathsf {M} _ {\varepsilon} \| \tilde {f} _ {x _ {i}} (x ; M , \psi)} \\ + 2 R _ {d} C q _ {\eta} e ^ {- \frac {c \sigma}{h} \log \frac {c \sigma}{h}} \sum_ {i = 1} ^ {d} \sum_ {j = 1} ^ {d} \sqrt {\| \mathsf {M} _ {\varepsilon} \| \tilde {f} _ {x _ {j} x _ {i}} (x ; M _ {\varepsilon} , \psi)}. \\ \end{array}
$$
It is clear that some big constant  $Q$ ,  $\tilde{f}_{x_j x_i}(x; \mathsf{M}_\varepsilon, \eta) \leq Q \| \mathsf{M}_\varepsilon \|$ , for all  $i, j \in \{1, \dots, q\}$ . Similarly,  $\tilde{f}_{x_i}(x; \mathsf{M}_\varepsilon, \eta) \leq Q \| \mathsf{M}_\varepsilon \|$  for all  $i \in \{1, \dots, q\}$  and  $\tilde{f}_t(x; \mathsf{M}_\varepsilon, \eta) \leq Q \| \mathsf{M}_\varepsilon \|$ . Hence, we have,
$$
\left| \frac {\partial (\hat {p} (x , t) - \tilde {p} (x , t))}{\partial t} + \sum_ {i = i} ^ {d} \frac {\partial (\mu_ {i} (x , t) (\hat {p} (x , t) - \tilde {p} (x , t)))}{\partial x _ {i}} - \sum_ {i = 1} ^ {d} \sum_ {j = 1} ^ {d} D _ {i j} \frac {\partial^ {2}}{\partial x _ {i} \partial x _ {j}} (\hat {p} (x, t) - \tilde {p} (x, t)) \right|
$$
$$
\begin{array}{l} \leq \left(2 + d R _ {\mu} + 2 d R _ {\mu} + 4 d ^ {2} R _ {d}\right) \| \mathsf {M} _ {\varepsilon} \| C ^ {2} q _ {\eta} ^ {2} e ^ {- \frac {2 c \sigma}{h} \log \frac {c \sigma}{h}} + \left(2 + d R _ {\mu_ {p}} + 2 d R _ {\mu} + 2 d ^ {2} R _ {d}\right) \sqrt {\| \mathsf {M} _ {\varepsilon} \| p (x , t)} C q _ {\eta} e ^ {- \frac {c \sigma}{h} \log \frac {c \sigma}{h}} \\ + 2 Q C q _ {\eta} e ^ {- \frac {c \sigma}{h} \log \frac {c \sigma}{h}} \| \mathsf {M} _ {\varepsilon} \| + (2 Q R _ {\mu} d + 4 Q d ^ {2} R _ {d}) C q _ {\eta} e ^ {- \frac {c \sigma}{h} \log \frac {c \sigma}{h}} \| \mathsf {M} _ {\varepsilon} \| + 2 Q R _ {d} d ^ {2} C q _ {\eta} e ^ {- \frac {c \sigma}{h} \log \frac {c \sigma}{h}} \| \mathsf {M} _ {\varepsilon} \| \\ \end{array}
$$
$$
\begin{array}{l} \leq (2 + d R _ {\mu} + 2 d R _ {\mu} + 4 d ^ {2} R _ {d}) \| \mathsf {M} _ {\varepsilon} \| C ^ {2} q _ {\eta} ^ {2} e ^ {- \frac {2 c \sigma}{h} \log \frac {c \sigma}{h}} + (2 + d R _ {\mu_ {p}} + 2 d R _ {\mu} + 2 d ^ {2} R _ {d}) \sqrt {\| \mathsf {M} _ {\varepsilon} \| p (x , t)} C q _ {\eta} e ^ {- \frac {c \sigma}{h} \log \frac {c \sigma}{h}} \\ + Q (2 C + 2 R _ {\mu} d + 6 Q d ^ {2} R _ {d}) C q _ {\eta} e ^ {- \frac {c \sigma}{h} \log \frac {c \sigma}{h}} \| \mathbf {M} _ {\varepsilon} \|. \\ \end{array}
$$
Now, let us compute  $\| \mathsf{M}_{\varepsilon}\|$ . We have
$$
\left\| \mathsf {M} _ {\varepsilon} \right\| \leq T r (M _ {\varepsilon}) = \hat {C} \tau^ {(d + 1) / 2} \left(1 + \varepsilon^ {\frac {2 \beta}{\beta - 2}} e ^ {\frac {\tilde {C} ^ {\prime}}{\tau} \varepsilon^ {- \frac {2}{\beta - 2}}}\right).
$$
Let us choose,  $\tau = \frac{\tilde{C}^{\prime}\varepsilon^{-\frac{2}{\beta - 2}}}{\frac{2\beta}{\beta - 2}\log(\frac{1 + R}{\varepsilon})}$ . Hence,
$$
\left\| M _ {\varepsilon} \right\| \leq C _ {3} (1 + R) ^ {\frac {2 \beta}{\beta - 2}} \varepsilon^ {- \frac {d + 1}{\beta - 2}}.
$$
$$
\left\| \frac {\partial (\hat {p} (x , t) - \tilde {p} (x , t))}{\partial t} + \sum_ {i = i} ^ {d} \frac {\partial (\mu_ {i} (x , t) (\hat {p} (x , t) - \tilde {p} (x , t)))}{\partial x _ {i}} - \sum_ {i = 1} ^ {d} \sum_ {j = 1} ^ {d} D _ {i j} \frac {\partial^ {2}}{\partial x _ {i} \partial x _ {j}} (\hat {p} (x, t) - \tilde {p} (x, t)) \right\| _ {L ^ {2} (\tilde {\mathfrak {X}})}
$$
$$
\begin{array}{l} \leq \left(2 + 3 d R _ {\mu} + 4 d ^ {2} R _ {d}\right) \| \mathsf {M} _ {\varepsilon} \| C ^ {2} q _ {\eta} ^ {2} e ^ {- \frac {2 c \sigma}{h} \log \frac {c \sigma}{h}} R ^ {(d + 1) / 2} + Q (2 C + 2 R _ {\mu} d + 6 Q d ^ {2} R _ {d}) C q _ {\eta} e ^ {- \frac {c \sigma}{h} \log \frac {c \sigma}{h}} \| \mathsf {M} _ {\varepsilon} \| R ^ {(d + 1) / 2} \\ + \left(2 + d R _ {\mu_ {p}} + 2 d R _ {\mu} + 2 d ^ {2} R _ {d}\right) \sqrt {\| M _ {\varepsilon} \| p (x , t)} C q _ {\eta} e ^ {- \frac {c \sigma}{h} \log \frac {c \sigma}{h}} \\ \end{array}
$$
$$
\begin{array}{l} = \underbrace{\left(2 + 3dR_{\mu} + 4d^{2}R_{d}\right)}_{:= P_{1}}\| \mathsf{M}_{\varepsilon}\| C^{2}q_{\eta}^{2}e^{\frac{-2c\sigma}{h}\log \frac{c\sigma}{h}}R^{(d + 1) / 2} + \underbrace{Q(2C + 2R_{\mu}d + 6Qd^{2}R_{d})}_{:= P_{2}}Cq_{\eta}e^{-\frac{c\sigma}{h}\log \frac{c\sigma}{h}}\| \mathsf{M}_{\varepsilon}\| R^{(d + 1) / 2} \\ + \underbrace {\left(2 + 3 d R _ {\mu} + 2 d ^ {2} R _ {d}\right)} _ {:= P _ {3}} \| \sqrt {\| \mathsf {M} _ {\varepsilon} \| p (x , t)} \| _ {L ^ {2} (\tilde {(\mathcal {X})})} C q _ {\eta} e ^ {- \frac {c \sigma}{h} \log \frac {c \sigma}{h}} \\ = P _ {1} \| \mathsf {M} _ {\varepsilon} \| C ^ {2} q _ {\eta} ^ {2} e ^ {- \frac {2 c \sigma}{h} \log \frac {c \sigma}{h}} R ^ {(d + 1) / 2} + P _ {2} C q _ {\eta} e ^ {- \frac {c \sigma}{h} \log \frac {c \sigma}{h}} \| \mathsf {M} _ {\varepsilon} \| R ^ {(d + 1) / 2} \\ + P _ {3} \| \mathbb {M} _ {\varepsilon} \| ^ {1 / 2} C q _ {\eta} e ^ {- \frac {c \sigma}{h} \log \frac {c \sigma}{h}} \| \sqrt {p (x , t)} \| _ {L ^ {2} (\tilde {\mathfrak {X}})}. \tag {51} \\ \end{array}
$$
Then, note that  $\| p^{1 / 2}\|_{L^2 (\mathcal{X})} = \| p\|_{L^1 (\mathcal{X})}^{1 / 2}$ . Hence, using the argument in [RC21], we have
$$
\left\| p ^ {1 / 2} \right\| _ {L ^ {2} (\tilde {x})} \leq 2 R.
$$
Term  $R$  comes from the fact that for each time instance  $p(x,t)$  is a density and  $t\in (0 < R)$ . Hence,
$$
\begin{array}{l} \left\| \frac {\partial (\hat {p} (x , t) - \tilde {p} (x , t))}{\partial t} + \sum_ {i = i} ^ {d} \frac {\partial (\mu_ {i} (x , t) (\hat {p} (x , t) - \tilde {p} (x , t)))}{\partial x _ {i}} - \sum_ {i = 1} ^ {d} \sum_ {j = 1} ^ {d} D _ {i j} \frac {\partial^ {2}}{\partial x _ {i} \partial x _ {j}} (\hat {p} (x, t) - \tilde {p} (x, t)) \right\| _ {L ^ {2} (\tilde {\mathfrak {X}})} \\ \leq P _ {1} \| \mathsf {M} _ {\varepsilon} \| C ^ {2} q _ {\eta} ^ {2} e ^ {- \frac {2 c \sigma}{h} \log \frac {c \sigma}{h}} R ^ {(d + 1) / 2} + P _ {2} C q _ {\eta} e ^ {- \frac {c \sigma}{h} \log \frac {c \sigma}{h}} \| \mathsf {M} _ {\varepsilon} \| R ^ {(d + 1) / 2} \\ + 2 P _ {3} R \| \mathsf {M} _ {\varepsilon} \| ^ {1 / 2} C q _ {\eta} e ^ {- \frac {c \sigma}{h} \log \frac {c \sigma}{h}} \\ \leq P _ {1} C _ {3} (1 + R) ^ {\frac {2 \beta}{\beta - 2}} \varepsilon^ {- \frac {d + 1}{\beta - 2}} C ^ {2} q _ {\eta} ^ {2} e ^ {- \frac {2 c \sigma}{h} \log \frac {c \sigma}{h}} R ^ {(d + 1) / 2} + P _ {2} C q _ {\eta} e ^ {- \frac {c \sigma}{h} \log \frac {c \sigma}{h}} C _ {3} (1 + R) ^ {\frac {2 \beta}{\beta - 2}} \varepsilon^ {- \frac {d + 1}{\beta - 2}} R ^ {(d + 1) / 2}. \\ + 2 P _ {3} R C _ {3} ^ {1 / 2} (1 + R) ^ {\frac {\beta}{\beta - 2}} \varepsilon^ {- \frac {d + 1}{2 (\beta - 2)}} C q _ {\eta} e ^ {- \frac {c \sigma}{h} \log \frac {c \sigma}{h}}. \\ \end{array}
$$
By choosing  $h = c\sigma / s$  with  $s = \max(C', (1 + \frac{d + 1}{2(\beta - 2)})\log \frac{1}{\varepsilon} + (1 + \frac{d}{2})\log(1 + R) + \log(\hat{C}) + e)$ , for some big enough  $\hat{C}$ . Since  $s \geq e$ , then  $\log s \geq 1$ , so
$$
C e ^ {- \frac {c \sigma}{h} \log \frac {c \sigma}{h}} = C e ^ {- s \log s} \leq C e ^ {- s} \leq C ^ {\prime} (1 + R) ^ {- d / 2} \varepsilon^ {1 + \frac {d + 1}{2 (\beta - 2)}}.
$$
Hence, we have,
$$
\begin{array}{l} \left\| \frac {\partial (\hat {p} (x , t) - \tilde {p} (x , t))}{\partial t} + \sum_ {i = i} ^ {d} \frac {\partial (\mu_ {i} (x , t) (\hat {p} (x , t) - \tilde {p} (x , t)))}{\partial x _ {i}} - \sum_ {i = 1} ^ {d} \sum_ {j = 1} ^ {d} D _ {i j} \frac {\partial^ {2}}{\partial x _ {i} \partial x _ {j}} (\hat {p} (x, t) - \tilde {p} (x, t)) \right\| _ {L ^ {2} (\hat {X})} \\ = O (\varepsilon). \\ \end{array}
$$
Combining everything together and by the use of triangles inequality, we have
$$
\left\| \frac {\partial \hat {p} (x , t)}{\partial t} + \sum_ {i = 1} ^ {d} \frac {\partial}{\partial x _ {i}} (\mu_ {i} (x, t) \hat {p} (x, t)) - \sum_ {i = 1} ^ {d} \sum_ {j = 1} ^ {d} D _ {i j} \frac {\partial^ {2}}{\partial x _ {i} \partial x _ {j}} \hat {p} (x, t) \right\| _ {L ^ {2} (\tilde {X})} = O (\varepsilon).
$$
To conclude we recall the fact that  $\tilde{x}_1, \ldots, \tilde{x}_m$  is a  $h$ -covering of  $T$ , guarantees that the number of centers  $m$  in the covering satisfies
$$
m \leq (1 + \frac {2 R \sqrt {d + 1}}{h}) ^ {d + 1}.
$$
Then, since  $h \geq c\sigma / (C_4 \log \frac{C_5 \log(1 + R)}{\varepsilon})$  with  $C_4 = 1 + d / \min(2(\beta - 2), 2)$  and  $C_5 = (\hat{C})^{1/C_4}$  and since  $\sigma = \min(R, 1/\sqrt{\tau})$ , then  $R / \sigma = \max(1, R\sqrt{\tau}) \leq 1 + \sqrt{\tilde{C}'} \varepsilon^{-1/(\beta - 2)} (\log \frac{1 + R}{\varepsilon})^{-1/2}$ , so after final calculation, we have
$$
m ^ {\frac {1}{d + 1}} \leq 1 + 2 R \sqrt {d + 1} / h \leq C _ {8} + C _ {9} \log \frac {1 + R}{\varepsilon} + C _ {1 0} \varepsilon^ {- \frac {1}{\beta - 2}} \left(\log \frac {1 + R}{\varepsilon}\right) ^ {1 / 2},
$$
for some constants  $C_8, C_9$  and  $C_{10}$  independent of  $\varepsilon$ .
A.7 Proof of Corollary 1
Proof of Corollary 1. Let  $p^{\star}$  be the probability corresponding to the solution of the Fokker-Planck equation with drift  $\mu$  and  $\hat{p}$  be the conditional probability corresponding to the solution of the Fokker-Planck equation with drift  $\hat{\mu}$ . And denote by  $\hat{p}_{\gamma}(x,t)$  the  $m$  dimensional PSD model obtained by applying the construction of Thm. 2 to the Fokker-Planck equation defined in terms of  $\hat{\mu}$ . Hence, from the direct application of the result in [BRS16, Theorem 1.1] since  $\| \mu - \hat{\mu} \|_{L^2} = O(\varepsilon)$  we have,
$$
\left\| p ^ {\star} - \hat {p} \right\| _ {L ^ {2} (T \times X)} = O (\varepsilon). \tag {52}
$$
From, theorem 2, we have,
$$
\| \hat {p} - \hat {p} _ {\gamma} \| _ {L ^ {2} (T \times X)} = O (\varepsilon).
$$
Finally due to proposition 2, for any PSD model  $f$  there exist an algorithm that samples i.i.d. from the probability  $p_f$  satisfying
$$
\mathbb {W} _ {1} (f, p _ {f}) = O (\varepsilon).
$$
Now note that for any  $t$ , the function  $\hat{p}_{\gamma}(t,\cdot)$  is still a PSD model (see [RC21]). Then, for any  $t\in (0,T)$ , we have that there exists probabilities  $\hat{p}_{t,\alpha}$  such that
$$
\mathbb {W} _ {1} \left(\hat {p} _ {\gamma} (t, \cdot), \hat {p} _ {t, \alpha} (\cdot)\right) = O (\varepsilon).
$$
Then, since  $\mathcal{W}_1$  is a distance and is bounded by the  $L^2$  distance since we are on a bounded domain, there exists a  $C$  depending on the diameter and the volume of the domain, such that for any  $t$
$$
\mathbb {W} _ {1} (p ^ {\star} (t, \cdot), \hat {p} _ {t, \alpha} (\cdot)) ^ {2} \leq C (\| p ^ {\star} (t, \cdot) - \hat {p} ^ {\star} (t, \cdot) \| _ {L ^ {2} (X)} ^ {2} + \| \hat {p} (t, \cdot) - \hat {p} _ {\gamma} (t, \cdot) \| _ {L ^ {2} (X)} ^ {2} + \mathbb {W} _ {1} (\hat {p} _ {\gamma}, p _ {\alpha}) ^ {2})
$$
Then, by taking the expectation over  $t\in (0,T)$  we have
$$
\mathbb {E} _ {t} \mathbb {W} _ {1} (p ^ {\star} (t, \cdot), \hat {p} _ {t, \alpha} (\cdot)) ^ {2} \leq C (\| p ^ {\star} - \hat {p} ^ {\star} \| _ {L ^ {2} (T \times X)} ^ {2} + \| \hat {p} (t, \cdot) - \hat {p} _ {\gamma} (t, \cdot) \| _ {L ^ {2} (T \times X)} ^ {2} + \mathbb {W} _ {1} (\hat {p} _ {\gamma}, p _ {\alpha}) ^ {2}) = O (\varepsilon^ {2})
$$
B Estimation of Fractional Laplacian Operator on a Probability Density
Before we go into the details of the estimation, let us first recall bochner's theorem.
Theorem 9 ([Rud17]). A continuous kernel  $k(x,y) = k(x - y)$  on  $\mathbb{R}^d$  is positive definite if and only if  $k(\delta)$  is the Fourier transform of a non-negative measure.
Proof of Theorem 3. In our psd model approximation, we have
$$
p (x) = f (x; A, X, \eta) = \sum_ {i, j = 1} ^ {m} A _ {i j} k (x _ {i}, x) k (x _ {j}, x).
$$
We will first prove the part (i) of the result. Now, we know that, fractional laplacian operator is defined as,
$$
\begin{array}{l} \mathcal {F} [ (- \Delta) ^ {s} f ] (\xi) = \| \xi \| ^ {2 s} \mathcal {F} [ f ] (\xi) \\ = \| \xi \| ^ {2 s} \sum_ {i, j = 1} ^ {m} A _ {i j} \mathcal {F} [ k (x _ {i}, \cdot) k (x _ {j}, \cdot) ] (\xi) \\ = \| \xi \| ^ {2 s} \sum_ {i, j} ^ {m} A _ {i j} \mathcal {F} [ k (x _ {i}, \cdot) ] (\xi) * \mathcal {F} [ k (x _ {j}, \cdot) ] (\xi). \\ \end{array}
$$
Now that,  $k$  is a gaussian kernel with bandwidth  $\eta$ . Hence,
$$
\begin{array}{l} k (y, x) = e ^ {- \eta \| y - x \| _ {2} ^ {2}} \\ \mathcal {F} [ k (y, \cdot) ] (\xi) = c e ^ {j \xi^ {\top} y - \frac {1}{4 \eta} \| \xi \| _ {2} ^ {2}} \\ = c e ^ {- \frac {1}{4 \eta} \left(\| \xi \| _ {2} ^ {2} - 4 j \eta \xi^ {\top} y - 4 \eta^ {2} \| y \| ^ {2} + 4 \eta^ {2} \| y \| ^ {2}\right)} \\ = c e ^ {- \eta \| y \| ^ {2}} \cdot e ^ {- \frac {1}{4 \eta} \| \xi - 2 j \eta y \| ^ {2}}, \\ \end{array}
$$
for some  $c > 0$  which can be computed in closed form. Now, we know that convolution of two multivariate gaussian functions are still gaussian. Hence,
$$
\mathcal {F} [ k (x _ {i}, \cdot) ] (\xi) * \mathcal {F} [ k (x _ {j}, \cdot) ] (\xi) = c ^ {\prime} e ^ {- \eta \left(\| x _ {i} \| ^ {2} + \| x _ {j} \| ^ {2}\right)} e ^ {- \frac {1}{8 \eta} \| \xi - 2 j \eta (x _ {i} + x _ {j}) \| ^ {2}},
$$
for some constant  $c' > 0$  which again can be computed in the closed form. Hence, we have
$$
\begin{array}{l} \mathcal {F} [ (- \Delta) ^ {s} f ] (\xi) = \| \xi \| ^ {2 s} \sum_ {i, j = 1} ^ {m} A _ {i j} \mathcal {F} [ k (x _ {i}, \cdot) ] (\xi) * \mathcal {F} [ k (x _ {j}, \cdot) ] (\xi) \\ = c ^ {\prime} \sum_ {i, j = 1} ^ {m} A _ {i j} \| \xi \| ^ {2 s} e ^ {- \eta \left(\| x _ {i} \| ^ {2} + \| x _ {j} \| ^ {2}\right)} e ^ {- \frac {1}{8 \eta} \| \xi - 2 j \eta \left(x _ {i} + x _ {j}\right) \| ^ {2}}. \tag {53} \\ \end{array}
$$
Finally we have,
$$
\begin{array}{l} (- \Delta) ^ {s} f (x) = C ^ {\prime} \sum_ {i, j = 1} ^ {m} A _ {i j} \int \| \xi \| ^ {2 s} e ^ {- \eta \left(\| x _ {i} \| ^ {2} + \| x _ {j} \| ^ {2}\right)} e ^ {- \frac {1}{8 \eta} \| \xi - 2 j \eta \left(x _ {i} + x _ {j}\right) \| ^ {2}} e ^ {- j \xi^ {\top} x} d \xi \\ = C ^ {\prime} \sum_ {i, j = 1} ^ {m} A _ {i j} e ^ {- \eta (\| x _ {i} \| ^ {2} + \| x _ {j} \| ^ {2})} \int \| \xi \| ^ {2 s} e ^ {- j \xi^ {\top} x} \cdot e ^ {- \frac {1}{8 \eta} \| \xi - 2 j \eta (x _ {i} + x _ {j}) \| ^ {2}} d \xi . \\ \end{array}
$$
By inspecting the above equation carefully, we can see that after appropriate scaling,
$$
(- \Delta) ^ {s} f (x) = C \sum_ {i, j = 1} ^ {m} A _ {i j} e ^ {- \eta \left(\| x _ {i} \| ^ {2} + \| x _ {j} \| ^ {2}\right)} \mathbb {E} _ {\xi \sim \mathcal {N} (\mu , \Sigma)} \left[ \| \xi \| ^ {2 s} e ^ {- j \xi^ {\top} x} \right] \tag {54}
$$
where  $\mu = 2j\eta (x_i + x_j)$  and  $\Sigma = 4\eta I$ . One can sample from  $\mathcal{N}(\mu, \Sigma)$  to estimate the fractional laplacian operator for PSD model.
Now, we will prove part (ii) of the theorem statement. From the bochner's theorem, we have,
$$
k (x - y) = \int_ {\mathbb {R} ^ {d}} q (\omega) e ^ {j \omega^ {\top} (x - y)} d \omega ,
$$
where  $q$  is the probability density. In our psd model approximation, we have
$$
\begin{array}{l} p (x) = f (x; A, X, \eta) = \sum_ {i, j = 1} ^ {m} A _ {i j} k (x _ {i}, x) k (x _ {j}, x) \\ = \sum_ {i, j = 1} ^ {m} A _ {i j} \left(\int_ {\mathbb {R} ^ {d}} e ^ {j \omega_ {\ell} ^ {\top} (x _ {i} - x)} q (\omega_ {\ell}) d \omega_ {\ell}\right) \left(\int_ {\mathbb {R} ^ {d}} e ^ {j \omega_ {k} ^ {\top} (x _ {j} - x)} q (\omega_ {k}) d \omega_ {k}\right) \\ = \sum_ {i, j = 1} ^ {m} A _ {i j} \int_ {\mathbb {R} ^ {d}} \int_ {\mathbb {R} ^ {d}} e ^ {j \omega_ {\ell} ^ {\top} (x _ {i} - x)} e ^ {j \omega_ {k} ^ {\top} (x _ {j} - x)} q (\omega_ {\ell}) q (\omega_ {k}) d \omega_ {\ell} d \omega_ {k}. \\ \end{array}
$$
Now, we know that, fractional laplacian operator is defined as,