def test_brentsroot(): for fmt in D.available_float_fmt(): print("Set dtype to:", fmt) D.set_float_fmt(fmt) for _ in range(10): ac_prod = D.array(np.random.uniform(0.9, 1.1)) a = D.array(np.random.uniform(-1, 1)) a = D.to_float(-1 * (a <= 0) + 1 * (a > 0)) c = ac_prod / a b = D.sqrt(0.01 + 4 * ac_prod) gt_root = -b / (2 * a) - 0.1 / (2 * a) ub = -b / (2 * a) lb = -b / (2 * a) - 1.0 / (2 * a) fun = lambda x: a * x**2 + b * x + c assert (D.to_numpy(D.to_float(D.abs(fun(gt_root)))) <= 32 * D.epsilon()) root, success = de.utilities.optimizer.brentsroot(fun, [lb, ub], 4 * D.epsilon(), verbose=True) assert (success) assert (np.allclose(D.to_numpy(D.to_float(gt_root)), D.to_numpy(D.to_float(root)), 32 * D.epsilon(), 32 * D.epsilon())) assert (D.to_numpy(D.to_float(D.abs(fun(root)))) <= 32 * D.epsilon())
def test_newtonraphson_pytorch_jacobian(ffmt, tol): print("Set dtype to:", ffmt) D.set_float_fmt(ffmt) np.random.seed(21) if tol is not None: tol = tol * D.epsilon() if D.backend() == 'torch': import torch torch.set_printoptions(precision=17) torch.autograd.set_detect_anomaly(False) if ffmt == 'gdual_vdouble': pytest.skip("Root-finding is ill-conceived with vectorised gduals") for _ in range(10): ac_prod = D.array(np.random.uniform(0.9, 1.1)) a = D.array(np.random.uniform(-1, 1)) a = D.to_float(-1 * (a <= 0) + 1 * (a > 0)) c = ac_prod / a b = D.sqrt(0.01 + 4 * ac_prod) gt_root1 = -b / (2 * a) - 0.1 / (2 * a) gt_root2 = -b / (2 * a) + 0.1 / (2 * a) ub = -b / (2 * a) - 0.2 / (2 * a) lb = -b / (2 * a) - 0.4 / (2 * a) x0 = D.array(np.random.uniform(ub, lb)) fun = lambda x: a * x**2 + b * x + c assert (D.to_numpy(D.to_float(D.abs(fun(gt_root1)))) <= 32 * D.epsilon()) assert (D.to_numpy(D.to_float(D.abs(fun(gt_root2)))) <= 32 * D.epsilon()) root, (success, num_iter, prec) = de.utilities.optimizer.newtonraphson(fun, x0, tol=tol, verbose=True) if tol is None: tol = D.epsilon() conv_root1 = np.allclose(D.to_numpy(D.to_float(gt_root1)), D.to_numpy(D.to_float(root)), 128 * tol, 32 * tol) conv_root2 = np.allclose(D.to_numpy(D.to_float(gt_root2)), D.to_numpy(D.to_float(root)), 128 * tol, 32 * tol) print(conv_root1, conv_root2, root, gt_root1, gt_root2, x0, root - gt_root1, root - gt_root2, num_iter, prec) assert (success) assert (conv_root1 or conv_root2) assert (D.to_numpy(D.to_float(D.abs(fun(root)))) <= 32 * tol)
def test_integration_and_nearestfloat_no_dense_output(): for ffmt in D.available_float_fmt(): D.set_float_fmt(ffmt) print("Testing {} float format".format(D.float_fmt())) de_mat = D.array([[0.0, 1.0],[-1.0, 0.0]]) @de.rhs_prettifier("""[vx, -x+t]""") def rhs(t, state, k, **kwargs): return de_mat @ state + D.array([0.0, t]) def analytic_soln(t, initial_conditions): c1 = initial_conditions[0] c2 = initial_conditions[1] - 1 return D.stack([ c2 * D.sin(D.to_float(D.asarray(t))) + c1 * D.cos(D.to_float(D.asarray(t))) + D.asarray(t), c2 * D.cos(D.to_float(D.asarray(t))) - c1 * D.sin(D.to_float(D.asarray(t))) + 1 ]) y_init = D.array([1., 0.]) a = de.OdeSystem(rhs, y0=y_init, dense_output=False, t=(0, 2*D.pi), dt=0.01, rtol=D.epsilon()**0.5, atol=D.epsilon()**0.5, constants=dict(k=1.0)) assert(a.integration_status() == "Integration has not been run.") a.integrate() assert(a.integration_status() == "Integration completed successfully.") assert(D.abs(a.t[-2] - a[2*D.pi].t) <= D.abs(a.dt))
def test_float_formats_typical_shape(ffmt, integrator, use_richardson_extrapolation, device): if use_richardson_extrapolation and integrator.__implicit__: pytest.skip( "Richardson Extrapolation is too slow with implicit methods") D.set_float_fmt(ffmt) if D.backend() == 'torch': import torch torch.set_printoptions(precision=17) torch.autograd.set_detect_anomaly(False) # Enable if a test fails device = torch.device(device) print("Testing {} float format".format(D.float_fmt())) from .common import set_up_basic_system de_mat, rhs, analytic_soln, y_init, dt, _ = set_up_basic_system( integrator, hook_jacobian=True) y_init = D.array([1., 0.]) if D.backend() == 'torch': y_init = y_init.to(device) a = de.OdeSystem(rhs, y0=y_init, dense_output=False, t=(0, D.pi / 4), dt=D.pi / 64, rtol=D.epsilon()**0.5, atol=D.epsilon()**0.5) method = integrator method_tolerance = a.atol * 10 + D.epsilon() if use_richardson_extrapolation: method = de.integrators.generate_richardson_integrator(method) method_tolerance = method_tolerance * 5 with de.utilities.BlockTimer(section_label="Integrator Tests") as sttimer: a.set_method(method) print("Testing {} with dt = {:.4e}".format(a.integrator, a.dt)) a.integrate(eta=True) print("Average step-size:", D.mean(D.abs(D.array(a.t[1:]) - D.array(a.t[:-1])))) max_diff = D.max(D.abs(analytic_soln(a.t[-1], y_init) - a.y[-1])) if a.integrator.adaptive: assert max_diff <= method_tolerance, "{} Failed with max_diff from analytical solution = {}".format( a.integrator, max_diff) if a.integrator.__implicit__: assert rhs.analytic_jacobian_called and a.njev > 0, "Analytic jacobian was called as part of integration" a.reset() print("") print("{} backend test passed successfully!".format(D.backend()))
def test_integration_and_representation(): for ffmt in D.available_float_fmt(): D.set_float_fmt(ffmt) print("Testing {} float format".format(D.float_fmt())) de_mat = D.array([[0.0, 1.0], [-1.0, 0.0]]) @de.rhs_prettifier("""[vx, -x+t]""") def rhs(t, state, k, **kwargs): return de_mat @ state + D.array([0.0, t]) def analytic_soln(t, initial_conditions): c1 = initial_conditions[0] c2 = initial_conditions[1] - 1 return D.stack([ c2 * D.sin(D.to_float(D.asarray(t))) + c1 * D.cos(D.to_float(D.asarray(t))) + D.asarray(t), c2 * D.cos(D.to_float(D.asarray(t))) - c1 * D.sin(D.to_float(D.asarray(t))) + 1 ]) def kbinterrupt_cb(ode_sys): if ode_sys[-1][0] > D.pi: raise KeyboardInterrupt("Test Interruption and Catching") y_init = D.array([1., 0.]) a = de.OdeSystem(rhs, y0=y_init, dense_output=True, t=(0, 2 * D.pi), dt=0.01, rtol=D.epsilon()**0.5, atol=D.epsilon()**0.5, constants=dict(k=1.0)) a.integrate() try: print(str(a)) print(repr(a)) assert (D.max(D.abs(a.sol(a.t[0]) - y_init)) <= 8 * D.epsilon()**0.5) assert (D.max( D.abs(a.sol(a.t[-1]) - analytic_soln(a.t[-1], y_init))) <= 8 * D.epsilon()**0.5) assert (D.max(D.abs(a.sol(a.t).T - analytic_soln(a.t, y_init))) <= 8 * D.epsilon()**0.5) except: raise
def test_matrix_inv(): A = D.array([ [-1.0, 3 / 2], [1.0, -1.0], ], dtype=D.float64) Ainv = D.matrix_inv(A) assert (D.max(D.abs(D.to_float(Ainv @ A - D.eye(2)))) <= 8 * D.epsilon())
def test_brentsroot_same_sign(): if D.backend() == 'torch': import torch torch.set_printoptions(precision=17) torch.autograd.set_detect_anomaly(True) ac_prod = D.array(np.random.uniform(0.9, 1.1)) a = D.array(1.0) b = D.array(1.0) gt_root = -b / a lb, ub = -b / a - 1, -b / a - 2 fun = lambda x: a * x + b assert (D.to_numpy(D.to_float(D.abs(fun(gt_root)))) <= 32 * D.epsilon()) root, success = de.utilities.optimizer.brentsroot(fun, [lb, ub], 4 * D.epsilon(), verbose=True) assert (np.isinf(root)) assert (not success)
def test_brentsroot_wrong_order(): if D.backend() == 'torch': import torch torch.set_printoptions(precision=17) torch.autograd.set_detect_anomaly(True) a = D.array(1.0) b = D.array(1.0) gt_root = -b / a lb, ub = -b / a - 1, -b / a + 1 fun = lambda x: a * x + b assert (D.to_numpy(D.to_float(D.abs(fun(gt_root)))) <= 32 * D.epsilon()) root, success = de.utilities.optimizer.brentsroot(fun, [ub, lb], 4 * D.epsilon(), verbose=True) assert (success) assert (np.allclose(D.to_numpy(D.to_float(gt_root)), D.to_numpy(D.to_float(root)), 32 * D.epsilon(), 32 * D.epsilon()))
def test_brentsrootvec(ffmt, tol): print("Set dtype to:", ffmt) D.set_float_fmt(ffmt) if tol is not None: tol = tol * D.epsilon() if D.backend() == 'torch': import torch torch.set_printoptions(precision=17) torch.autograd.set_detect_anomaly(True) if ffmt == 'gdual_vdouble': pytest.skip("Root-finding is ill-conceived with vectorised gduals") for _ in range(10): slope_list = D.array( np.copysign(np.random.uniform(0.9, 1.1, size=25), np.random.uniform(-1, 1, size=25))) intercept_list = slope_list gt_root_list = -intercept_list / slope_list fun_list = [(lambda m, b: lambda x: m * x + b)(m, b) for m, b in zip(slope_list, intercept_list)] assert (all( map((lambda i: D.to_numpy(D.to_float(D.abs(i))) <= 32 * D.epsilon( )), map((lambda x: x[0](x[1])), zip(fun_list, gt_root_list))))) root_list, success = de.utilities.optimizer.brentsrootvec( fun_list, [D.min(gt_root_list) - 1., D.max(gt_root_list) + 1.], tol, verbose=True) assert (np.all(D.to_numpy(success))) assert (np.allclose(D.to_numpy(D.to_float(gt_root_list)), D.to_numpy(D.to_float(root_list)), 32 * D.epsilon(), 32 * D.epsilon())) assert (all( map((lambda i: D.to_numpy(D.to_float(D.abs(i))) <= 32 * D.epsilon( )), map((lambda x: x[0](x[1])), zip(fun_list, root_list)))))
def test_integration_and_representation(): for ffmt in D.available_float_fmt(): D.set_float_fmt(ffmt) print("Testing {} float format".format(D.float_fmt())) de_mat = D.array([[0.0, 1.0],[-1.0, 0.0]]) @de.rhs_prettifier("""[vx, -x+t]""") def rhs(t, state, k, **kwargs): return de_mat @ state + D.array([0.0, t]) def analytic_soln(t, initial_conditions): c1 = initial_conditions[0] c2 = initial_conditions[1] - 1 return D.stack([ c2 * D.sin(D.to_float(D.asarray(t))) + c1 * D.cos(D.to_float(D.asarray(t))) + D.asarray(t), c2 * D.cos(D.to_float(D.asarray(t))) - c1 * D.sin(D.to_float(D.asarray(t))) + 1 ]) y_init = D.array([1., 0.]) a = de.OdeSystem(rhs, y0=y_init, dense_output=True, t=(0, 2*D.pi), dt=0.01, rtol=D.epsilon()**0.5, atol=D.epsilon()**0.5, constants=dict(k=1.0)) assert(a.integration_status() == "Integration has not been run.") a.integrate() assert(a.integration_status() == "Integration completed successfully.") try: print(str(a)) print(repr(a)) assert(D.max(D.abs(a.sol(a.t[0]) - y_init)) <= 8*D.epsilon()**0.5) assert(D.max(D.abs(a.sol(a.t[-1]) - analytic_soln(a.t[-1], y_init))) <= 8*D.epsilon()**0.5) assert(D.max(D.abs(a.sol(a.t).T - analytic_soln(a.t, y_init))) <= 8*D.epsilon()**0.5) except: raise for i in a: assert(D.max(D.abs(i.y - analytic_soln(i.t, y_init))) <= 8*D.epsilon()**0.5) assert(len(a.y) == len(a)) assert(len(a.t) == len(a))
def test_brentsroot(ffmt, tol): print("Set dtype to:", ffmt) D.set_float_fmt(ffmt) if tol is not None: tol = tol * D.epsilon() if D.backend() == 'torch': import torch torch.set_printoptions(precision=17) torch.autograd.set_detect_anomaly(True) for _ in range(10): ac_prod = D.array(np.random.uniform(0.9, 1.1)) a = D.array(np.random.uniform(-1, 1)) a = D.to_float(-1 * (a <= 0) + 1 * (a > 0)) c = ac_prod / a b = D.sqrt(0.01 + 4 * ac_prod) gt_root = -b / (2 * a) - 0.1 / (2 * a) ub = -b / (2 * a) lb = -b / (2 * a) - 1.0 / (2 * a) fun = lambda x: a * x**2 + b * x + c assert (D.to_numpy(D.to_float(D.abs(fun(gt_root)))) <= 32 * D.epsilon()) root, success = de.utilities.optimizer.brentsroot(fun, [lb, ub], tol, verbose=True) assert (success) assert (np.allclose(D.to_numpy(D.to_float(gt_root)), D.to_numpy(D.to_float(root)), 32 * D.epsilon(), 32 * D.epsilon())) assert (D.to_numpy(D.to_float(D.abs(fun(root)))) <= 32 * D.epsilon())
def test_event_detection(): for ffmt in D.available_float_fmt(): if ffmt == 'float16': continue D.set_float_fmt(ffmt) print("Testing event detection for float format {}".format(D.float_fmt())) de_mat = D.array([[0.0, 1.0],[-1.0, 0.0]]) @de.rhs_prettifier("""[vx, -x+t]""") def rhs(t, state, **kwargs): return de_mat @ state + D.array([0.0, t]) def analytic_soln(t, initial_conditions): c1 = initial_conditions[0] c2 = initial_conditions[1] - 1 return D.array([ c2 * D.sin(t) + c1 * D.cos(t) + t, c2 * D.cos(t) - c1 * D.sin(t) + 1 ]) y_init = D.array([1., 0.]) def time_event(t, y, **kwargs): return t - D.pi/8 time_event.is_terminal = True time_event.direction = 0 a = de.OdeSystem(rhs, y0=y_init, dense_output=True, t=(0, D.pi/4), dt=0.01, rtol=D.epsilon()**0.5, atol=D.epsilon()**0.5) with de.utilities.BlockTimer(section_label="Integrator Tests") as sttimer: for i in sorted(set(de.available_methods(False).values()), key=lambda x:x.__name__): try: a.set_method(i) print("Testing {}".format(a.integrator)) a.integrate(eta=True, events=time_event) if D.abs(a.t[-1] - D.pi/8) > 10*D.epsilon(): print("Event detection with integrator {} failed with t[-1] = {}".format(a.integrator, a.t[-1])) raise RuntimeError("Failed to detect event for integrator {}".format(str(i))) else: print("Event detection with integrator {} succeeded with t[-1] = {}".format(a.integrator, a.t[-1])) a.reset() except Exception as e: raise e raise RuntimeError("Test failed for integration method: {}".format(a.integrator)) print("") print("{} backend test passed successfully!".format(D.backend()))
def test_integration_and_nearest_float_no_dense_output(ffmt): D.set_float_fmt(ffmt) if D.backend() == 'torch': import torch torch.set_printoptions(precision=17) torch.autograd.set_detect_anomaly(True) print("Testing {} float format".format(D.float_fmt())) de_mat = D.array([[0.0, 1.0], [-1.0, 0.0]]) @de.rhs_prettifier("""[vx, -x+t]""") def rhs(t, state, k, **kwargs): return de_mat @ state + D.array([0.0, t]) y_init = D.array([1., 0.]) a = de.OdeSystem(rhs, y0=y_init, dense_output=False, t=(0, 2 * D.pi), dt=0.01, rtol=D.epsilon()**0.5, atol=D.epsilon()**0.5, constants=dict(k=1.0)) assert (a.integration_status == "Integration has not been run.") a.integrate() assert (a.sol is None) assert (a.integration_status == "Integration completed successfully.") assert (D.abs(a.t[-2] - a[2 * D.pi].t) <= D.abs(a.dt))
def test_matrix_inv_bigger(): for diag_size in range(2, 101): np.random.seed(15) for trial in range(3): A = np.random.normal(size=(diag_size, diag_size)) while np.abs(np.linalg.det(D.to_float(A))) <= 1e-5: A = np.random.normal(size=(diag_size, diag_size), std=250.0) A = D.array(D.cast_to_float_fmt(A)) Ainv = D.matrix_inv(A) assert ( D.max(D.abs(D.to_float(Ainv @ A - D.eye(diag_size)))) <= 4 * D.epsilon()**0.5 ), "Matrix inversion failed for diagonal with size: " + str( diag_size)
def test_brentsrootvec(): for fmt in D.available_float_fmt(): print("Set dtype to:", fmt) D.set_float_fmt(fmt) if fmt == 'gdual_vdouble': continue for _ in range(10): slope_list = D.array( np.copysign(np.random.uniform(0.9, 1.1, size=25), np.random.uniform(-1, 1, size=25))) intercept_list = slope_list gt_root_list = -intercept_list / slope_list fun_list = [(lambda m, b: lambda x: m * x + b)(m, b) for m, b in zip(slope_list, intercept_list)] assert (all( map((lambda i: D.to_numpy(D.to_float(D.abs(i))) <= 32 * D. epsilon()), map((lambda x: x[0](x[1])), zip(fun_list, gt_root_list))))) root_list, success = de.utilities.optimizer.brentsrootvec( fun_list, [D.min(gt_root_list) - 1., D.max(gt_root_list) + 1.], 4 * D.epsilon(), verbose=True) assert (np.all(D.to_numpy(success))) assert (np.allclose(D.to_numpy(D.to_float(gt_root_list)), D.to_numpy(D.to_float(root_list)), 32 * D.epsilon(), 32 * D.epsilon())) assert (all( map((lambda i: D.to_numpy(D.to_float(D.abs(i))) <= 32 * D. epsilon()), map((lambda x: x[0](x[1])), zip(fun_list, root_list)))))
def test_integration_and_representation(ffmt): D.set_float_fmt(ffmt) if D.backend() == 'torch': import torch torch.set_printoptions(precision=17) torch.autograd.set_detect_anomaly(True) print("Testing {} float format".format(D.float_fmt())) from . import common (de_mat, rhs, analytic_soln, y_init, dt, a) = common.set_up_basic_system() assert (a.integration_status == "Integration has not been run.") a.integrate() assert (a.integration_status == "Integration completed successfully.") print(str(a)) print(repr(a)) assert (D.max(D.abs(a.sol(a.t[0]) - y_init)) <= 8 * D.epsilon()**0.5) assert (D.max(D.abs(a.sol(a.t[-1]) - analytic_soln(a.t[-1], y_init))) <= 8 * D.epsilon()**0.5) assert (D.max(D.abs(a.sol(a.t).T - analytic_soln(a.t, y_init))) <= 8 * D.epsilon()**0.5) for i in a: assert (D.max(D.abs(i.y - analytic_soln(i.t, y_init))) <= 8 * D.epsilon()**0.5) assert (len(a.y) == len(a)) assert (len(a.t) == len(a))
def test_dense_output(ffmt, use_richardson_extrapolation): D.set_float_fmt(ffmt) if D.backend() == 'torch': import torch torch.set_printoptions(precision=17) torch.autograd.set_detect_anomaly(True) print("Testing {} float format".format(D.float_fmt())) from . import common (de_mat, rhs, analytic_soln, y_init, dt, a) = common.set_up_basic_system() assert (a.integration_status == "Integration has not been run.") if use_richardson_extrapolation: a.method = de.integrators.generate_richardson_integrator(a.method) a.rtol = a.atol = D.epsilon()**0.75 a.integrate() assert (a.integration_status == "Integration completed successfully.") assert (D.max(D.abs(a[0].y - analytic_soln(a[0].t, y_init))) <= 4 * D.epsilon()) assert (D.max(D.abs(a[0].t)) <= 4 * D.epsilon()) assert (D.max(D.abs(a[-1].y - analytic_soln(a[-1].t, y_init))) <= 10 * D.epsilon()**0.5) assert (D.max(D.abs(a[a[0].t].y - analytic_soln(a[0].t, y_init))) <= 4 * D.epsilon()) assert (D.max(D.abs(a[a[0].t].t)) <= 4 * D.epsilon()) assert (D.max(D.abs(a[a[-1].t].y - analytic_soln(a[-1].t, y_init))) <= 10 * D.epsilon()**0.5) assert (D.max(D.abs(D.stack(a[a[0].t:a[-1].t].y) - D.stack(a.y))) <= 4 * D.epsilon()) assert (D.max(D.abs(D.stack(a[:a[-1].t].y) - D.stack(a.y))) <= 4 * D.epsilon()) assert (D.max(D.abs(D.stack(a[a[0].t:a[-1].t:2].y) - D.stack(a.y[::2]))) <= 4 * D.epsilon()) assert (D.max(D.abs(D.stack(a[a[0].t::2].y) - D.stack(a.y[::2]))) <= 4 * D.epsilon()) assert (D.max(D.abs(D.stack(a[:a[-1].t:2].y) - D.stack(a.y[::2]))) <= 4 * D.epsilon()) np.random.seed(42) sample_points = D.array(np.random.uniform(a.t[0], a.t[-1], 1024)) assert (D.max( D.abs(a.sol(sample_points) - analytic_soln(sample_points, y_init).T)).item() <= D.epsilon()** 0.5)
def test_float_formats(): for ffmt in D.available_float_fmt(): D.set_float_fmt(ffmt) print("Testing {} float format".format(D.float_fmt())) de_mat = D.array([[0.0, 1.0], [-1.0, 0.0]]) @de.rhs_prettifier("""[vx, -x+t]""") def rhs(t, state, **kwargs): return de_mat @ state + D.array([0.0, t]) def analytic_soln(t, initial_conditions): c1 = initial_conditions[0] c2 = initial_conditions[1] - 1 return D.array([ c2 * D.sin(t) + c1 * D.cos(t) + t, c2 * D.cos(t) - c1 * D.sin(t) + 1 ]) def kbinterrupt_cb(ode_sys): if ode_sys[-1][0] > D.pi: raise KeyboardInterrupt("Test Interruption and Catching") y_init = D.array([1., 0.]) a = de.OdeSystem(rhs, y0=y_init, dense_output=True, t=(0, 2 * D.pi), dt=0.01, rtol=D.epsilon()**0.5, atol=D.epsilon()**0.5) with de.utilities.BlockTimer( section_label="Integrator Tests") as sttimer: for i in sorted(set(de.available_methods(False).values()), key=lambda x: x.__name__): if "Heun-Euler" in i.__name__ and D.float_fmt( ) == "gdual_real128": print( "skipping {} due to ridiculous timestep requirements.". format(i)) continue try: a.set_method(i) print("Testing {}".format(a.integrator)) try: a.integrate(callback=kbinterrupt_cb, eta=True) except KeyboardInterrupt as e: pass try: a.integrate(eta=True) except: raise max_diff = D.max( D.abs(analytic_soln(a.t[-1], a.y[0]) - a.y[-1])) if a.method.__adaptive__ and max_diff >= a.atol * 10 + D.epsilon( ): print( "{} Failed with max_diff from analytical solution = {}" .format(a.integrator, max_diff)) raise RuntimeError( "Failed to meet tolerances for adaptive integrator {}" .format(str(i))) else: print( "{} Succeeded with max_diff from analytical solution = {}" .format(a.integrator, max_diff)) a.reset() except Exception as e: print(e) raise RuntimeError( "Test failed for integration method: {}".format( a.integrator)) print("") print("{} backend test passed successfully!".format(D.backend()))
@de.rhs_prettifier( equ_repr="[vx, -k*x/m]", md_repr=r""" $$ \frac{dx}{dt} = \begin{bmatrix} 0 & 1 \\ -\frac{k}{m} & 0 \end{bmatrix} \cdot \begin{bmatrix}x \\ v_x\end{bmatrix} $$ """ ) def rhs(t, state, k, m, **kwargs): return D.array([[0.0, 1.0], [-k/m, 0.0]])@state y_init = D.array([1., 0.]) a = de.OdeSystem(rhs, y0=y_init, dense_output=True, t=(0, 2*D.pi), dt=0.01, rtol=1e-9, atol=1e-9, constants=dict(k=1.0, m=1.0)) print(a) a.integrate() print(a) print("If the integration was successful and correct, a[0].y and a[-1].y should be near identical.") print("a[0].y = {}".format(a[0].y)) print("a[-1].y = {}".format(a[-1].y)) print("Maximum difference from initial state after one oscillation cycle: {}".format(D.max(D.abs(a[0].y-a[-1].y))))
def test_tanh_within_tolerance(ffmt): D.set_float_fmt(ffmt) pi = D.to_float(D.pi) assert (D.abs(D.tanh(pi) - np.tanh(pi)) <= 2 * D.epsilon())
def test_abs_within_tolerance(ffmt): D.set_float_fmt(ffmt) pi = D.to_float(D.pi) assert (3.141592653589793 - 2 * D.epsilon() <= D.abs(pi) <= 3.141592653589793 + 2 * D.epsilon())
def test_atan2_within_tolerance(ffmt): D.set_float_fmt(ffmt) ref = D.to_float(D.pi / 4) assert (D.abs(D.atan2(D.to_float(1), D.to_float(1)) - ref) <= 2 * D.epsilon())
def test_asin_within_tolerance(ffmt): D.set_float_fmt(ffmt) ref = D.to_float(D.pi / 2) assert (D.abs(D.asin(D.to_float(1)) - ref) <= 2 * D.epsilon())
def test_acos_within_tolerance(ffmt): D.set_float_fmt(ffmt) assert (D.abs(D.acos(D.to_float(1))) <= 2 * D.epsilon())
def do(self, A, b): x = D.solve_linear_system(A, b) assert (D.max(D.abs(D.to_float(A @ x - b))) <= 256 * D.epsilon())
def test_gradients_simple_decay(ffmt, integrator, use_richardson_extrapolation, device): if use_richardson_extrapolation and integrator.__implicit__: pytest.skip( "Richardson Extrapolation is too slow with implicit methods") D.set_float_fmt(ffmt) if integrator.__symplectic__: pytest.skip( "Exponential decay system is not in the form compatible with symplectic integrators" ) print("Testing {} float format".format(D.float_fmt())) import torch torch.set_printoptions(precision=17) device = torch.device(device) torch.autograd.set_detect_anomaly(False) # Enable if a test fails def rhs(t, state, k, **kwargs): return -k * state def rhs_jac(t, state, k, **kwargs): return -k rhs.jac = rhs_jac y_init = D.array(5.0, requires_grad=True) csts = dict(k=D.array(1.0, device=device)) dt = max(0.5 * (D.epsilon()**0.5)**(1.0 / (max(2, integrator.order - 1))), 5e-2) def true_solution_decay(t, initial_state, k): return initial_state * D.exp(-k * t) method = integrator if use_richardson_extrapolation: method = de.integrators.generate_richardson_integrator(method) with de.utilities.BlockTimer(section_label="Integrator Tests"): y_init = D.ones((1, ), requires_grad=True).to(device) y_init = y_init * D.e a = de.OdeSystem(rhs, y_init, t=(0, 1.0), dt=dt, rtol=D.epsilon()**0.5, atol=D.epsilon()**0.5, constants=csts) a.set_method(method) print("Testing {} with dt = {:.4e}".format(a.integrator, a.dt)) a.integrate(eta=True) Jy = D.jacobian(a.y[-1], a.y[0]) true_Jy = D.jacobian(true_solution_decay(a.t[-1], a.y[0], **csts), a.y[0]) print(a.y[-1], true_solution_decay(a.t[-1], a.y[0], **csts), D.abs(a.y[-1] - true_solution_decay(a.t[-1], a.y[0], **csts))) print(a.integrator.adaptive, D.mean(D.abs(D.stack(a.t[1:]) - D.stack(a.t[:-1]))), D.norm(true_Jy - Jy), 32 * a.rtol) print(a.integrator.adaptive, true_Jy, Jy) if a.integrator.adaptive: assert (D.allclose(true_Jy, Jy, rtol=32 * a.rtol, atol=32 * a.atol)) print("{} method test succeeded!".format(a.integrator)) print("") print("{} backend test passed successfully!".format(D.backend()))
def test_gradients_simple_oscillator(ffmt, integrator, use_richardson_extrapolation, device): if use_richardson_extrapolation and integrator.__implicit__: pytest.skip( "Richardson Extrapolation is too slow with implicit methods") D.set_float_fmt(ffmt) print("Testing {} float format".format(D.float_fmt())) import torch torch.set_printoptions(precision=17) device = torch.device(device) torch.autograd.set_detect_anomaly(False) # Enable if a test fails def rhs(t, state, k, m, **kwargs): return D.array([[0.0, 1.0], [-k / m, 0.0]], device=device) @ state csts = dict(k=1.0, m=1.0) T = 2 * D.pi * D.sqrt(D.array(csts['m'] / csts['k'])).to(device) dt = max(0.5 * (D.epsilon()**0.5)**(1.0 / (max(2, integrator.order - 1))), 5e-2) def true_solution_sho(t, initial_state, k, m): w2 = D.array(k / m).to(device) w = D.sqrt(w2) A = D.sqrt(initial_state[0]**2 + initial_state[1]**2 / w2) phi = D.atan2(-initial_state[1], w * initial_state[0]) return D.stack([A * D.cos(w * t + phi), -w * A * D.sin(w * t + phi)]).T method = integrator if use_richardson_extrapolation: method = de.integrators.generate_richardson_integrator(method) with de.utilities.BlockTimer(section_label="Integrator Tests"): y_init = D.array([1., 1.], requires_grad=True).to(device) a = de.OdeSystem(rhs, y_init, t=(0, T), dt=T * dt, rtol=D.epsilon()**0.5, atol=D.epsilon()**0.5, constants=csts) a.set_method(method) print("Testing {} with dt = {:.4e}".format(a.integrator, a.dt)) a.integrate(eta=True) Jy = D.jacobian(a.y[-1], a.y[0]) true_Jy = D.jacobian(true_solution_sho(a.t[-1], a.y[0], **csts), a.y[0]) print(a.integrator.adaptive, D.mean(D.abs(D.stack(a.t[1:]) - D.stack(a.t[:-1]))), D.norm(true_Jy - Jy), D.epsilon()**0.5) if a.integrator.adaptive: assert (D.allclose(true_Jy, Jy, rtol=4 * a.rtol**0.75, atol=4 * a.atol**0.75)) print("{} method test succeeded!".format(a.integrator)) print("") print("{} backend test passed successfully!".format(D.backend()))
def do(self, A): Ainv = D.matrix_inv(A) assert (D.max(D.abs(D.to_float(Ainv @ A - D.eye(A.shape[0])))) <= 256 * D.epsilon())
def test_float_formats_atypical_shape(ffmt, integrator, use_richardson_extrapolation, device): if use_richardson_extrapolation and integrator.__implicit__: pytest.skip( "Richardson Extrapolation is too slow with implicit methods") D.set_float_fmt(ffmt) if D.backend() == 'torch': import torch torch.set_printoptions(precision=17) torch.autograd.set_detect_anomaly(False) # Enable if a test fails device = torch.device(device) print("Testing {} float format".format(D.float_fmt())) from .common import set_up_basic_system de_mat, _, analytic_soln, y_init, dt, _ = set_up_basic_system(integrator) @de.rhs_prettifier("""[vx, -x+t]""") def rhs(t, state, **kwargs): nonlocal de_mat extra = D.array([0.0, t]) if D.backend() == 'torch': de_mat = de_mat.to(state.device) extra = extra.to(state.device) return D.sum(de_mat[:, :, None, None, None] * state, axis=1) + extra[:, None, None, None] y_init = D.array([[[[1., 0.]] * 1] * 1] * 3).T print(rhs(0.0, y_init).shape) if D.backend() == 'torch': y_init = y_init.contiguous().to(device) a = de.OdeSystem(rhs, y0=y_init, dense_output=False, t=(0, D.pi / 4), dt=D.pi / 64, rtol=D.epsilon()**0.5, atol=D.epsilon()**0.5) method = integrator method_tolerance = a.atol * 10 + D.epsilon() if use_richardson_extrapolation: method = de.integrators.generate_richardson_integrator(method) method_tolerance = method_tolerance * 5 with de.utilities.BlockTimer(section_label="Integrator Tests") as sttimer: a.set_method(method) print("Testing {} with dt = {:.4e}".format(a.integrator, a.dt)) a.integrate(eta=True) max_diff = D.max(D.abs(analytic_soln(a.t[-1], y_init) - a.y[-1])) if a.integrator.adaptive: assert max_diff <= method_tolerance, "{} Failed with max_diff from analytical solution = {}".format( a.integrator, max_diff) a.reset() print("") print("{} backend test passed successfully!".format(D.backend()))
def do(self, A): Q, R = D.qr(A) assert (D.max(D.abs(D.to_float(Q @ R - A))) <= 256 * D.epsilon())