Results

Result for the paper “Towards Efficient Modelling of String Dynamics: A Comparison of State Space and Koopman Methods”

Animated Predictions

Non-linear Gaussian

Video 1

Video 1 shows the evolution of the non-linear string at 16kHz with Gaussian-like initial conditions. The first 300 steps are shown.

Linear Noise

Video 2

Video 2 shows the evolution of the linear string at 16kHz with noise-like initial conditions. Only the first 150 steps are shown.

Audio

The following table shows the audio predictions for a single point in the string for DMD, Koopman, KoopmanVAR, LRU, and S5 methods for both the linear noise and non-linear Gaussian initial conditions.

Linear Noise Non-linear Gaussian
Ground Truth
DMD
Koopman
KoopmanVAR
LRU
S5

Table of Results

Results for the non-linear dataset

Model Gaussian 4000kHz MSE Rel Gaussian 4000kHz MAE Rel Gaussian 16000kHz MSE Rel Gaussian 16000kHz MAE Rel Noise 4000kHz MSE Rel Noise 4000kHz MAE Rel Noise 16000kHz MSE Rel Noise 16000kHz MAE Rel
DMD 1.1865(0.2019) 0.8289(0.0857) 1.0938(0.1311) 0.8747(0.0749) 1.6695(0.2687) 1.0420(0.1178) 1.8313(0.2467) 1.0953(0.087)
Koopman 0.1041(0.0144) 0.3271(0.0259) 0.0971(0.0029) 0.3117(0.0044) 0.2070(0.0939) 0.3978(0.1084) 0.0865(0.0155) 0.2596(0.0275)
Koopman_var 0.0113(0.0061) 0.0977(0.0284) 0.0094(0.0046) 0.0888(0.0245) 0.0041(0.0008) 0.0527(0.0081) 0.0531(0.0572) 0.1714(0.1056)
LRU 0.0260(0.0067) 0.1571(0.0240) 0.0390(0.0187) 0.1870(0.0450) 0.0250(0.0065) 0.1284(0.0077) 0.0476(0.0058) 0.1922(0.0125)
S5 0.1165(0.0260) 0.3318(0.0291) 0.0224(0.0013) 0.1467(0.0053) 0.0507(0.0127) 0.1916(0.0190) 0.0387(0.0062) 0.1493(0.0078)

Results for the linear dataset

Model Gaussian 4000kHz MSE Rel Gaussian 4000kHz MAE Rel Gaussian 16000kHz MSE Rel Gaussian 16000kHz MAE Rel Noise 4000kHz MSE Rel Noise 4000kHz MAE Rel Noise 16000kHz MSE Rel Noise 16000kHz MAE Rel
DMD 0.0000(0.0000) 0.0060(0.0007) 0.0001(0.0000) 0.0097(0.0006) 0.0000(0.0000) 0.0042(0.0004) 0.0001(0.0000) 0.0066(0.0004)
Koopman 0.0110(0.0069) 0.1005(0.0331) 0.0048(0.0009) 0.0692(0.0074) 0.0073(0.0052) 0.0789(0.0322) 0.0019(0.0001) 0.0423(0.0004)
Koopman_var 0.0020(0.0013) 0.0427(0.0145) 0.0013(0.0012) 0.0321(0.0142) 0.0002(0.0001) 0.0134(0.0023) 0.0495(0.0973) 0.1125(0.1785)
LRU 0.0014(0.0005) 0.0373(0.0065) 0.0010(0.0002) 0.0317(0.0034) 0.0005(0.0001) 0.0199(0.0025) 0.0005(0.0002) 0.0217(0.0036)
S5 0.0386(0.0273) 0.1896(0.0597) 0.0156(0.0102) 0.1206(0.0421) 0.0112(0.0029) 0.1023(0.0139) 0.0037(0.0012) 0.0539(0.0119)

Mean and standard deviation (in parentheses) for 5 different seeds of the a) non-linear and b) linear validation data across different models and sampling rates, under Gaussian and noise-like initial conditions. We use 4000 steps for both 4kHz and 16kHz. Since DMD does not depend on a seed, we include the standard deviation of the MSE and MAE across the validation dataset.